Using central limit theorem, the probability that the class of 20 students will have a mean completion time greater than 60 minutes on the unit 5 test is 0.00017332
What is the probability that your class of 20 students will have a mean completion time greater than 60 minutes on the unit 5 test?We can use the Central Limit Theorem (CLT) to approximate the distribution of the sample mean completion time for the class. According to CLT, the distribution of the sample mean is approximately normal, with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.
In this case, the population mean is given as 48 minutes, the population standard deviation is given as 15 minutes, and the sample size is 20. Therefore, the mean of the sample mean completion time is also 48 minutes, and the standard deviation of the sample mean completion time is 15/√20 ≈ 3.3541 minutes.
To find the probability that the class mean completion time is greater than 60 minutes, we can standardize the distribution of the sample mean completion time using the z-score formula:
z = (x - μ) / (σ / √n)
where x is the value we want to find the probability for (in this case, x = 60), μ is the population mean, σ is the population standard deviation, and n is the sample size.
Plugging in the values, we get:
z = (60 - 48) / (15 / √20) = 3.5777
Using a standard normal distribution table (or calculator), we can find the probability that a z-score is greater than 3.5777.
P = 0.00017332
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A right triangle is describe as having an angle of measure six less than negative two times a number, another angle measure that is three less than negative one-fourth the number, and a right angle. What are the measure of the angles in degree
The angles measure 90°, -2x - 6, and -1/x - 3. ⇒ x = -44. Therefore, the required measures of the angles are 90°, 82°, and 8° in the given triangle.
A right triangle is a type of triangle where one of the angles measures exactly 90 degrees. This angle is known as the right angle, and it is formed by the intersection of the two sides of the triangle that are perpendicular to each other. The other two angles of the right triangle are acute angles, meaning they measure less than 90 degrees.
The side opposite the right angle is called the hypotenuse, and it is always the longest side of the right triangle. The other two sides are called legs, and they can be of different lengths. This theorem is one of the most important and useful tools in geometry, and it allows us to solve many practical problems involving right triangles, such as finding the height of a building.
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A 10 kilogram object suspended from the end of a vertically hanging spring stretches the spring 9.8 centimeters. At time t = 0, the resulting mass-spring system is disturbed from its rest state by the force F(t) = 100cos(8t). The force F(t) is expressed in Newtons and is positive in the downward direction, and time is measured in seconds.
Determine the spring constant k.
k = ? Newtons / meter
Formulate the initial value problem for y(t), where y(t) is the displacement of the object from its equilibrium rest state, measured positive in the downward direction. (Give your answer in terms of y,y′,y′′,t.)
Differential equation: ?
Initial conditions: y(0) = ? and y′(0) = ?
Solve the initial value problem for y(t).
y(t) = ?
Plot the solution and determine the maximum excursion from equilibrium made by the object on the time interval 0 ≤ t < [infinity]. If there is no such maximum, enter NONE.
maximum excursion = ? meters
Using Hooke's law the maximum value of |cos(8t)| is 1, so the maximum excursion is 1/6 meters.
To find the spring constant k, we use Hooke's law:
F = -ky
where F is the weight of the object, and y is the distance it is stretched from its rest position. At equilibrium, F = mg = 10 × 9.81 = 98.1 N. Thus,
98.1 = -k × 0.098
k = -1000 N/m
The equation of motion for the system is given by:
my'' + ky = F(t)
Substituting the given values, we get:
10y'' + (-1000)y = 100cos(8t)
y'' - 100y = 10cos(8t)
with initial conditions y(0) = 0 and y'(0) = 0.
The characteristic equation is r² - 100 = 0, with roots r = ±10i. The complementary solution is therefore y_c(t) = c1cos(10t) + c2sin(10t).
For the particular solution, we assume a form of yp(t) = Acos(8t) + Bsin(8t), and substitute it in the differential equation to get:
-64Acos(8t) - 64Bsin(8t) - 100(Acos(8t) + Bsin(8t)) = 10cos(8t)
Solving for A and B, we get A = -1/6 and B = 0. Thus, the particular solution is yp(t) = (-1/6) × cos(8t).
The general solution is therefore y(t) = c1cos(10t) + c2sin(10t) - (1/6)*cos(8t). Applying the initial conditions, we get c1 = 0 and c2 = 0, so the particular solution is simply y(t) = (-1/6) × cos(8t).
The maximum excursion from equilibrium can be found by taking the absolute value of y(t) and finding its maximum value. We have:
|y(t)| = (1/6) × |cos(8t)|
The maximum value of |cos(8t)| is 1, so the maximum excursion is 1/6 meters.
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a garrison has provision for 30 days for certain men. if 2/3 of them do not attend the mess, then the food will last for? (b) 65 days (d) none of
A garrison has provision for 30 days for certain men. if 2/3 of them do not attend the mess, then the food will last for 45 days. So, the correct option is (a).
How to calculateGiven that the provision for certain men in the garrison is for 30 days. Also, given that 2/3 of them do not attend the mess, then we have to find the number of days the food will last.
The food will last longer if the number of people attending the mess is less because the same amount of food will have to be shared between fewer people. Therefore, the food will last for more than 30 days.
Let the total number of men be x, then the number of men attending the mess is (1/3)x
And the number of men not attending the mess is (2/3)x.
Therefore, the food will last for (30 × x) / (2/3)x = 45 days
Hence, the answer of the question is 46 days.
Your question is incomplete but most probably your full question was:
A garrison has provision for 30 days for certain men. if 2/3 of them do not attend the mess, then the food will last for?
(a) 45 days
(b) 65 days
(c) 50 days
(d) none of above
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Kids's Kingdom, a retail toy chain, placed a seasonal order for stuffed animals from Stuffed Stuff, a distributor. Large animals cost $20, and small ones cost $14.
If the total cost of the order was $7,320 for 450 pieces, how many of each size were ordered? What was the dollar amount of each size ordered?
Answer:
Kids's Kingdom ordered 170 large stuffed animals and 280 small stuffed animals. The dollar amount of each size ordered was $3,400 for the large stuffed animals and $3,920 for the small stuffed animals.
Step-by-step explanation:
Let's use the following variables:
L for the number of large stuffed animals
S for the number of small stuffed animals
We can set up a system of two equations to represent the given information:
L + S = 450 (equation 1)
20L + 14S = 7320 (equation 2)
We can solve this system of equations using substitution or elimination. Let's use substitution.
From equation 1, we can solve for L:
L = 450 - S
Substitute this expression for L into equation 2:
20(450 - S) + 14S = 7320
Distribute the 20:
9000 - 20S + 14S = 7320
Simplify and solve for S:
6S = 1680
S = 280
So, Kids's Kingdom ordered 280 small stuffed animals. We can use equation 1 to find the number of large stuffed animals:
L + 280 = 450
L = 170
Therefore, Kids's Kingdom ordered 170 large stuffed animals.
To find the dollar amount of each size ordered, we can multiply the number of each size by the cost per item:
170 large stuffed animals at $20 each: 170 * $20 = $3,400
280 small stuffed animals at $14 each: 280 * $14 = $3,920
So, Kids's Kingdom spent $3,400 on large stuffed animals and $3,920 on small stuffed animals for a total cost of $7,320.
please help with question 6
Answer:
a = -13b = 6f(x) = (2x -1)(x -2)(x +3)Step-by-step explanation:
Given f(x) = 2x³ +x² +ax +b has a factor (x -2) and a remainder of 18 when divided by (x -1), you want to know a, b, and the factored form of f(x).
RemainderIf (x -2) is a factor, then the value of f(2) is zero:
f(2) = 2·2³ +2² +2a +b = 0
2a +b = -20 . . . . . . . subtract 20
If the remainder from division by (x +1) is 18, then f(-1) is 18:
f(-1) = 2·(-1)³ +(-1)² +a·(-1) +b = 18
-a +b = 19 . . . . . . . . . . add 1
Solve for a, bSubtracting the second equation from the first gives ...
(2a +b) -(-a +b) = (-20) -(19)
3a = -39
a = -13
b = 19 +a = 6
The values of 'a' and 'b' are -13 and 6, respectively.
Factored formWe can find the quadratic factor using synthetic division, given one root is x=2. The tableau for that is ...
[tex]\begin{array}{c|cccc}2&2&1&-13&6\\&&4&10&-6\\\cline{1-5}&2&5&-3&0\end{array}[/tex]
The remainder is 0, as expected, and the quadratic factor of f(x) is 2x² +5x -3. Now, we know f(x) = (x -2)(2x² +5x -3).
To factor the quadratic, we need to find factors of (2)(-3) = -6 that have a sum of 5. Those would be 6 and -1. This lets us factor the quadratic as ...
2x² +5x -3 = (2x +6)(2x -1)/2 = (x +3)(2x -1)
The factored form of f(x) is ...
f(x) = (2x -1)(x -2)(x +3)
HELP PLEASE.
The options of answer to this question are:
A- 286 miles
B- 271 miles
C- 166 miles
D- 216 miles
PLEASE HELP
Answer: D - 216 miles
Explanation: The rest stop is the y-intercept in the graph because that is when he begins to travel home at a constant speed. Looking at the graph, the y-intercept falls right before 220 so the closest answer would be 216 miles.
PLEASE HELP MARKING BRAINLEIST JUST ANSWER ASAP
Answer:
The pink triangle
Step-by-step explanation:
IMPORTANT NOTE: Make sure all the units are the same and consistent
Perimeter of a figure = Total length of the outer boundary
Shape of each figure in this question = Isosceles Triangle
Perimeter of triangle = Sum of all three sides
Perimeter of pink triangle = 36m + 36m + 20m
= 92m
Perimeter of green triangle = 25m + 25m + 35m
= 85m
∴Comparing the two values calculated above, it can be observed that the pink triangle has a greater perimeter
17 Troy scored 945 points playing 3 games of pinball. He scored
312 points in the first game and 356 points in the second game.
How many points did Troy score in the third game?
Answer:
Troy scored 277 points in the third game of pinball.
Step-by-step explanation:
Let x be the number of points Troy scored in the third game.
We know that Troy scored a total of 945 points in 3 games, so we can set up an equation:
312 + 356 + x = 945
Simplifying this equation, we get:
668 + x = 945
Subtracting 668 from both sides, we get:
x = 277
Therefore, Troy scored 277 points in the third game of pinball.
Help!
I need your help.
Consider a square whose side-length is one unit. Select any five points from inside this square. Prove that at least two of these points are within squareroot 2/2 units of each other.
The given square with a side length of one unit is known to contain five points. One must prove that at least two of these points are within square root 2/2 units of each other.
According to the Pigeonhole principle, "if n items are put into m containers, with n > m, then at least one container must contain more than one item."In this context, the square is the container, and the points inside it are the objects. If more than four points are picked, the theorem is true, and two points are nearer to each other than the square root of 2/2 units.
Let's place four points on the square's four corners. The distance between any two of these points is the square root of two units since the square's side length is 1.
Let's add another point to the mix. That point is either inside the square or outside it. Without loss of generality, let us assume that the point is inside the square. It must then be within the perimeter outlined by joining the square's corners to the point that was not a corner already.
The perimeter of the square described above is a square with a side length of square root 2 units.
Since we have five points in the square, at least two of them must be in the same smaller square, due to the pigeonhole principle. Without loss of generality, let's assume that two of the points are in the upper-left square. As a result, any points within this square are within the square root 2 units of any of the other four points. Hence, at least two points of the five selected are within the square root of 2/2 units of each other.
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Find the equation of the straight line passing through the point (0,2) which is perpendicular to the line y=1/4x+5
Answer:
y = -4x + 2
Step-by-step explanation:
you need to find the gradient first and in order to find it, you need to look at the equation of the line given
in the equation, it refers to y = mx + c and from there, the gradient is whatever the value of m is. So in this situation, m = 1/4
now that you've found your gradient, you need to get the gradient when it is perpendicular (as stated in the question) by using m¹ x m² = -1
m¹ represents the gradient of the line we have whereas m² represents the gradient of the line we want so you just have to substitute 1/4 into m¹
[tex] \frac{1}{4} \times {m}^{2} = - 1[/tex]
[tex] {m}^{2} = \frac{ - 1}{( \frac{1}{4} )} [/tex]
[tex] {m}^{2} = - 4[/tex]
now you need to find the c of the y = mx + c before you complete the equation
y = 2 (from the question)
x = 0 (from the question)
m = -4
(2) = (-4)(0) + c
2 = c
c = 2
and you just substitute everything except y into y = mx + c and you're done
y = -4x + 2
Find the missing side of the triangle below
The value of y in the given triangle is 7.44 units.
What is tangent in trigonometry?The trigonometric ratio between the opposing and adjacent sides of a right triangle that contains an angle is its tangent. The trigonometric functions, also known as circular, angle, or goniometric functions in mathematics, are real functions that link the angle of a right-angled triangle to the ratios of its two side lengths. All geosciences, including navigation, solid mechanics, celestial mechanics, geodesy, and many more, utilise them extensively.
The given triangle is a right-triangle.
The trigonometric identity that gives the relationship between opposite side and adjacent side is tan.
Thus,
tan (28) = opposite / adjacent = y / 14
y = 0.53 (14)
y = 7.44
Hence, the value of y in the given triangle is 7.44 units.
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You purchased 40 shares for $3.95/sh. If you sold the shares for a total of $200. Did you net a profit or a loss?
Answer: profit
Step-by-step explanation:
3.95 (price of one share) multiplied by 40 (the amount of shares bought) would have cost $158. so selling all for $200 would be a $42 profit
*9. The consultancy Imagination Inc. Is working with its manufacturing client Parts-R-Us to improve their on-time
performance. The firm can earn a bonus of up to $1,000,000 based on how much the on-time performance actually
improves. It's current (baseline) on-time performance is 90%.
The company typically completes approximately 1,000 orders per month, with approximately 100 orders delayed. The
bonus payment is prorated according to the following criteria:
• The on-time performance improvement is calculated based on a reduction in late events or an improvement in on-
time performance.
• No bonus is earned for the first 25% reduction in late events, say from 100 to 75. Maximum bonus is earned once
Parts-R-Us achieves 95% on-time performance.
Please answer the following:
Write down a formula to
determine the total bonus
amount to be received
Using your formula, show
how much bonus would be
paid if Parts-R-Us achieves
94% on-time performance
Parts-R-Us could be eligible for a $0 bonus if it achieves an on-time performance rate of 94% or more.
The company would have to maintain an order delivery rate of 95% in order to qualify for the bonus. If achieved, the bonus would be as follows: Bonus = 94% * (1 - 100).
The following is the calculation for the bonus:
Bonus = P * (1 - D)
Where:
P represents the proportion of orders that are delivered on time.
D represents the total number of orders placed late.
In the event that Parts-R-Us achieves an on-time performance rate of 94%, the bonus would be as follows:
Bonus = 94% * (1 - 100)
Bonus = $0
If the company only managed to complete its tasks 94% of the time, it will not be eligible for a bonus.
In order to qualify for the maximum bonus, Parts-R-Us would have to maintain an on-time delivery rate of 95%.
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find two positive real numbers such that the sum of the first number and the second number is 48 and their product is a maximum
Answer:
x = 24 and y = 24
Step-by-step explanation:
Let's use algebra to solve this optimization problem.
Let x be the first number, and y be the second number. Then we have the following two equations based on the problem statement:
x + y = 48 (sum of the two numbers is 48)
xy = ? (product of the two numbers, which we want to maximize)
To solve for x and y in terms of each other, we can use the fact that:
(x + y)^2 = x^2 + 2xy + y^2
Expanding the left side of the equation gives:
x^2 + 2xy + y^2 = 2304
And substituting xy for its value in terms of x and y gives:
x^2 + 2xy + y^2 = x^2 + 2(48 - x)y + y^2 = 2304
Simplifying this equation gives:
2y^2 - 96y + x^2 - 2304 = 0
To maximize the product xy, we need to maximize the value of xy = x(48 - x) = 48x - x^2. This function is a quadratic that opens downwards, and therefore, its maximum value occurs at the vertex of the parabola, which is located at x = -b/2a = -48/(2*-1) = 24.
Thus, the two positive real numbers that sum up to 48 and their product is a maximum are x = 24 and y = 24.
Can someone help me find the elevation of the sun I need the answers that are highlighted in yellow please help image below
Answer:
Step-by-step explanation:
a. ∠ACB
b. AC
c. AB
d. BC
e. tangent, opposite, adjacent
f. m∠ACB = tan⁻¹(34/45) = 37°
Two similar solids have base areas of 47 cm² and 199 cm², as shown below.
The volume of the smaller solid is 350 cm³.
COMPLETION
50%
Calculate the volume of the larger solid correct to the nearest integer.
(4 marks)
Check the picture below.
so hmmm let's use the ratio for the areas to get the ratio of the sides, and from there, we'll get to the ratio of the volumes.
[tex]\stackrel{ \textit{Areas' ratio} }{\sqrt{\cfrac{s^2}{s^2}}}=\cfrac{s}{s}\implies \sqrt{\cfrac{47}{199}}=\cfrac{s}{s}\implies \cfrac{\sqrt{47}}{\sqrt{199}}=\cfrac{s}{s} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\stackrel{ \textit{Volumes' ratio} }{\sqrt[3]{\cfrac{s^3}{s^3}}}=\cfrac{s}{s}\implies \stackrel{\textit{substituting from above}}{\sqrt[3]{\cfrac{s^3}{s^3}}=\cfrac{\sqrt{47}}{\sqrt{199}}}\implies \sqrt[3]{\cfrac{350}{V}}=\cfrac{\sqrt{47}}{\sqrt{199}} \\\\\\ \cfrac{350}{V}=\left( \cfrac{\sqrt{47}}{\sqrt{199}} \right)^3\implies \cfrac{350}{V}=\cfrac{\sqrt{47^3}}{\sqrt{199^3}}\implies (350)(\sqrt{199^3})=V\sqrt{47^3} \\\\\\ \cfrac{(350)(\sqrt{199^3})}{\sqrt{47^3}}=V\implies \boxed{3049\approx V}[/tex]
Fractions questions need help!
The answer to this question is 150 adults. This is calculated by subtracting the number of boys and girls from the total number of people in the museum, 250.
What is subtracting?Subtracting is a mathematical operation that involves the removal of one number or quantity from another. Subtracting can be done by either counting down from the larger number or counting up from the smaller number until the two numbers meet.
2/5 of 250 people is equal to 100 girls. 3/10 of 250 people is equal to 75 boys. When these two numbers are subtracted from the total number of people in the museum, 250, the answer is 150 adults.
To work out the number of adults in the museum, it is important to first identify the fractions and convert them into decimals. For example, to convert 2/5 into a decimal, 2 is divided by 5, which gives an answer of 0.4. This process should be repeated for the other fractions given in this problem.
Once the fractions are converted into decimals, the next step is to multiply the decimals by the total number of people in the museum, 250. For example, 0.4 multiplied by 250 is equal to 100 girls.
Finally, the numbers of boys and girls should be subtracted from the total number of people in the museum, 250. This gives an answer of 150 adults.
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By subtracting the number of boys and girls from the total number of people in the museum, we get the number of adults that is 75.
What is subtracting?
Subtracting is a mathematical operation that involves the removal of one number or quantity from another. Subtracting can be done by either counting down from the larger number or counting up from the smaller number until the two numbers meet.
2/5 of 250 people = 100 girls.
3/10 of 250 people =75 boys.
When these two numbers are subtracted from the total number of people in the museum, that is
250-(100+75)= 75 adults
Thus, the number of adults among the 250 people in a museum are 75.
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150,000 bonds with a coupon rate of 11 percent and a current price quote of 108; the bonds have 20 years to maturity. 320,000 zero coupon bonds with a price quote of 16 and 30 years until maturity. Both bonds have a par value of $1,000 and semiannual coupons
The total value of both bonds is $704,367,500.
Coupon payment = [tex]\frac{Coupon rate * Par value}{2}[/tex]
Coupon payment = [tex]\frac{11 * $1,000}{2}[/tex]
Coupon payment = $55
PV = [tex]55 * [1 - (1 + 0.04)^{^-40} ] / 0.04 + $1,000 / (1 + 0.04)^40[/tex]
[tex]PV = $1,026.45[/tex]
[tex]Total value = PV * Number of bonds * Par value\\Total value = $1,026.45 * 150,000 * $1,000\\Total value = $153,967,500[/tex]
[tex]PV = \frac{Price}{(1 + r)^n}[/tex]
[tex]PV = \frac{16}{(1 +0.03)^60}\\PV = $1.72[/tex]
[tex]Total value = PV * Number of bonds * Par value\\Total value = $1.72 * 320,000 * $1,000\\Total value = $550,400,000[/tex]
Therefore, the total value of both bonds is:
[tex]Total value = Value of coupon bonds + Value of zero coupon bonds\\Total value = $153,967,500 + $550,400,000\\Total value = $704,367,500[/tex]
A coupon rate is the annual interest rate paid by a bond or other fixed-income security to its bondholders or investors. It is typically expressed as a percentage of the bond's face value, also known as its par value. For example, if a bond has a face value of $1,000 and a coupon rate of 5%, the bond will pay $50 in interest each year to its bondholders. The coupon payments are usually made semi-annually or annually, depending on the terms of the bond.
The coupon rate is set when the bond is issued and remains fixed throughout the life of the bond unless the bond issuer chooses to call the bond or the bond defaults. Coupon rates are determined by a variety of factors, including market conditions, the creditworthiness of the issuer, and the length of the bond's maturity.
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Complete Question: -
The IPO Investment Bank has the following financing outstanding,
Debt: 150,000 bonds with a coupon rate of 11 percent and a current price quote of 108; the bonds have 20 years to maturity. 320,000 zero coupon bonds with a price quote of 16 and 30 years until maturity. Both bonds have a par value of $1,000 and semiannual coupons.
Preferred stock: 240,000 shares of 9 percent preferred stock with a current price of $67, and a par value of $100.
Common stock: 3,500,000 shares of common stock; the current price is $53, and the beta of the stock is.9.
Market: The corporate tax rate is 24 percent, the market risk premium is 8 percent, and the risk-free rate is 5 percent.
What is the WACC for the company? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)
Which explicit function defines this arithmetic sequence?
-351, -343, -335, -327, -319
The explicit function defines this arithmetic sequence:
C. f(n) = 8n − 359
Explicit Function:
An explicit function is a function expressed with arguments. For example, y = 4x – 7 is self-explanatory, where y is the dependent variable and depends on the independent variable x.
According to the Question:
The first element of the given arithmetic sequence is -351, and the tolerance is (-343 -(-351)) = 8. The tolerance is a multiplier of n in the explicit function.
We can identify the appropriate explicit function by finding the one that correctly describes the sequence. Evaluating each for n=1 is sufficient.
(A) f(1) = 8 -351 = -343
(B) f(1) = -8 -351 = -359
(C) f(1) = 8 -359 = -351
(D) f(1) = -8 +359 = 351
The sequence is defined by the explicit function :
f(n) = 8n -359
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1. Find the length of an arc of a circle with radius 21 m that subtends a central angle of 15°
The length of the arc is approximately 5.51 meters when a circle with a radius of 21 meters is subtended by a central angle of 15 degrees.
The length of an arc of a circle with radius 21m that subtends a central angle of 15° can be calculated using the formula:
Arc length = (central angle/360°) x 2πr
where r is the radius of the circle, and π is the mathematical constant pi.
Substituting the given values, we get:
Arc length = (15/360) x 2π x 21
Arc length = (1/24) x 2 x 3.14 x 21
Arc length = (1/12) x 3.14 x 21
Arc length = 5.51 meters (rounded to two decimal places).
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We need to apply the formula to determine the length of a circle's arc: (Central angle / 360°) x (2 x x radius) is the formula for arc length. the radius is distance from the circle center to any point on its perimeter,
and the central angle is the angle subtended by the arc at its center. The radius in this instance is stated as 21 meters, while the arc's center angle is provided as 15 degrees. When these values are added to the formula, we obtain: arc length is equal to (15°/360°) x (2x x 21m) 3.68 m. As a result, the arc measures around 3.68 meters in length. As a result, the radius is the distance from the circle's center to any point on its perimeter, if we were to sketch an arc of The arc's length would be around 3.68 meters for a circle with a radius of 21 meters and a center angle of 15 degrees.
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Kay borrows $8,750 from Chase Bank at a fixed annual interest rate of 6.2%,
where y is the number of years the loan remains unpaid. Kay waits 8 years
before beginning to repay the loan to Citibank.
Write an exponential function to represent the amount owed after t years.
O y = 8750(t) 6.2
Oy = 8750 (1.062)
Oy (t) 1.0628750
Answer:
Answer: $80,000(1+0.004)^60 + $519.17 [1-(+0.004)^60/0.004]
Step-by-step explanation:
let be the linear transformation given by let be the basis of given by and let be the basis of given by find the coordinate matrix of relative to the ordered bases and . HW6.7. Coordinate matrix for differentiation Let L :P2P be the linear transformation given by L(p(t)) = 5p"(t) + 1p' (t) + 3p(t) + 3tp(t). Let E = (C1, C2, C3) be the basis of P2 given by el(t) = 1, ez(t) = t, ez(t) = ť. and let F = (f1, 82, 83, fa) be the basis of P3 given by fi(t) = 1, fz(t) = t, fz(t) = {2, fa(t) = {'. Find the coordinate matrix LFE of L relative to the ordered bases & and F. LFE = Save & Grade 2 tries left Save only
The coordinate matrix LFE of L relative to the ordered bases E and F is
[tex]\left(\begin{array}{ccc}3&1&10\\3&3&2\\0&3&3\\0&0&3\end{array}\right)[/tex].
Since here L is a linear transformation from a vector space of dimension 3 to a vector space of dimension 4, the coordinate matrix of L relative to the given ordered bases must be a (4×3) matrix,
Linear transformation is given by,
L(P(t)) = 5p"(t) + 1p' (t) + 3p(t) + 3tp(t)
The given basis is IP² is E = (C1, C2, C3) where, e1(t) = 1, e2(t) = t, e3(t) = t².
Also the given basis of IP³ is (f1, f2, f3, f4) where, f1(t) = 1, f2(t) = t, f3(t) = t², f4(t) = t³.
Now to find the coordinate matrix,
Now,
L(e1(t)) = 5.0 + 1.0 + 3.1 + 3t.1
= 3 + 3t
= 3f1(t) + 3f2(t) + 0.f3(t) + 0.f4(t)
L(e2(t)) = 5.0 + 1.1 + 3.t + 3t.t
= 1 + 3t + 3t²
L(e3(t)) = 5.2 + 1.2t + 3.t² + 3t.t²
= 10 + 2t + 3t² + 3t³
Now writing the coefficients as a column vector we get,
[tex]\left(\begin{array}{ccc}3&1&10\\3&3&2\\0&3&3\\0&0&3\end{array}\right)[/tex]
The coordinate matrix LFE of L relative to the ordered bases E and F is
[tex]\left(\begin{array}{ccc}3&1&10\\3&3&2\\0&3&3\\0&0&3\end{array}\right)[/tex].
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Find the compound interest and the total amount after eight years if the interest is compounded every two years.
Principal = ₹10,000
Rate of interest = 20%
Total amount = (find)
Total interest = (find)
After 8 years, the total amount is ₹38,416 and the compound interest is ₹28,416.
What is the total amount and compound interest earned on ₹10,000 invested at 20% interest compounded every 2 years for 8 years?
To find the compound interest and the total amounts after eight year with interest compounded every two years, we'll use the compound interest formula:
Total Amount (A) = P(1 + r/n)¹/²(nt)
Where:
P = Principal = ₹10,000
r = Rate of interest = 20% = 0.2
n = Number of times the interest is compounded in a year (every 2 years, so n = 1/2)
t = Time in years = 8 years
Convert the interest rate to a decimal by dividing by 100:
20% ÷ 100 = 0.2
A = ₹10,000x (1 + 0.2¹/²)(1/2 x 8)
Calculate the expression inside the parentheses:
1 + 0.2/(1/2) = 1.4
Calculate the exponent (1/2x 8):
1/2x 8 = 4
Calculate the total amount:
A = ₹10,000 x (1.4)^4
A = ₹10,000 x3.8416
A = ₹38,416
Step 6: Calculate the compound interest:
Total interest = Total amount - Principal
Total interest = ₹38,416 - ₹10,000
Total interest = ₹28,416
So, after eight years, the total amount is ₹38,416 and the compound interest is ₹28,416.
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Express the following as the product of prime factors in exponential form
(a) 432 (b) 729×64
Answer: 729×64 is: (3^3 × 2^3)^2
Step-by-step explanation:
(a) To express 432 as the product of prime factors in exponential form, we can follow these steps:
Divide by 2 as many times as possible until the result is odd: 432 ÷ 2 = 216 ÷ 2 = 108 ÷ 2 = 54 ÷ 2 = 27 (5 times)
Divide by 3 as many times as possible until the result is not divisible by 3: 27 ÷ 3 = 9 ÷ 3 = 3 (2 times)
Since 3 is a prime number, we cannot divide by any other prime number to obtain a smaller result. Therefore, the prime factorization of 432 is: 2^4 × 3^3.
(b) To express 729×64 as the product of prime factors in exponential form, we can follow these steps:
Rewrite each factor as a power of a prime: 729 = 3^6 and 64 = 2^6.
Multiply the powers of each prime together: (3^6) × (2^6) = 3^6 × 2^6.
Simplify the result by factoring out the highest possible power of each prime: 3^6 × 2^6 = (3^3 × 2^3)^2.
Therefore, the prime factorization of 729×64 is: (3^3 × 2^3)^2.
Answer:
Below in bold.
Step-by-step explanation:
2) 432
2) 216
2) 108
2) 54
3) 27
3) 9
3
So 432 = 2^4 * 3^3.
3)729
3)243
3)81
3)27
3)9
3
64 = 2^6
So the answer is 2^6 * 3^6
At the book store, you purchased some $5 clearance mystery books and $12 regular-priced science fiction books. How many of each did you buy if you spent a total of $126?
Answer: View answer in explanation below.
Step-by-step explanation: Let's use variables to represent the unknown quantities.
Let x be the number of $5 clearance mystery books purchased.
Let y be the number of $12 regular-priced science fiction books purchased.
We can set up a system of equations based on the given information:
5x + 12y = 126 (total amount spent)
x + y = total number of books purchased
We need to solve for x and y.
Let's use the second equation to solve for one variable in terms of the other:
y = total number of books purchased - x
Now we can substitute this expression for y into the first equation:
5x + 12(total number of books purchased - x) = 126
Simplifying and solving for x:
5x + 12total number of books purchased - 12x = 126
-7x + 12total number of books purchased = 126
-7x = -12total number of books purchased + 126
x = (12total number of books purchased - 126)/7
Since x must be a whole number (you can't buy a fraction of a book), we need to find a value of total number of books purchased that makes x a whole number. We can start by trying different values of total number of books purchased:
If total number of books purchased is 10:
x = (12(10) - 126)/7 = -6/7 (not a whole number)
If total number of books purchased is 11:
x = (12(11) - 126)/7 = 6/7 (not a whole number)
If total number of books purchased is 12:
x = (12(12) - 126)/7 = 6/7 (not a whole number)
If total number of books purchased is 13:
x = (12(13) - 126)/7 = 12/7 (not a whole number)
If total number of books purchased is 14:
x = (12(14) - 126)/7 = 18/7 (not a whole number)
If total number of books purchased is 15:
x = (12(15) - 126)/7 = 24/7 (not a whole number)
If total number of books purchased is 16:
x = (12(16) - 126)/7 = 30/7 (not a whole number)
If total number of books purchased is 17:
x = (12(17) - 126)/7 = 36/7 (not a whole number)
If total number of books purchased is 18:
x = (12(18) - 126)/7 = 42/7 = 6 (a whole number)
So, you bought 6 $5 clearance mystery books and 12 - 6 = 6 $12 regular-priced science fiction books.
in a heptagon, the degree measures of the interior angles are $x, ~x, ~x-2, ~x-2, ~x 2, ~x 2$ and $x 4$ degrees. what is the degree measure of the largest interior angle?
The degree measure of the largest interior angle of the given heptagon is 132.57 degrees.
The largest interior angle of the given heptagon. The degree measures of the interior angles of a heptagon are 7, with 7 sides or vertices, are $x, ~x, ~x-2, ~x-2, ~x+2, ~x+2,$ and $x+4$ degrees.
The sum of the degree measures of the interior angles of a polygon with n sides is given by $S = (n-2) \c dot 180^\circ$. The sum of the interior angles of a heptagon is given by $S = (7-2) \cdot 180^\circ = 900^\circ$.
The sum of the degree measures of the interior angles of the heptagon is equal to $x+x+x-2+x-2+x+2+x+2+x+4 = 7x+4$. To find the value of x, we will set this equation equal to the total sum of the interior angles:$7x+4 = 900^\circ$. Solving for x, we get$x = 128.57$
We may now substitute the value of x to get the degree measures of each of the angles in the heptagon:$x = 128.57^\circ$$x = 128.57^\circ$$x - 2 = 126.57^\circ$$x - 2 = 126.57^\circ$$x + 2 = 130.57^\circ$$x + 2 = 130.57^\circ$$x + 4 = 132.57^\circ$
To find the degree measure of the largest interior angle, we must look for the angle with the largest value. We can see that the largest angle measures $132.57^\circ$.
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What x-values are a solution to the System?
Select EACH correct answer.
Question 2 options:
x = -2
x = -1
x = 0
x = 1
x = 2
The solutions to the system equations are x = -1 and x = 0.
What is a system of equations?When two or more variables are related to one another and equations are constructed to determine each variable's value, the result is a system of equations. an equation is a balance scale, and for the equation to stay true, both sides must be equal.
Given equations are:
[tex]y=2^x - 1\\y=\frac{1}{2} x[/tex]
The solutions of the system for which; [tex]y_{1} =y_{2}[/tex]
According to the given table, the solutions to the system of both equations at x = -1 and x = 0.
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seven less than the product of a number n and 1/4 is no more than 95
[tex]\frac{1}{4} n - 7 \leq 95[/tex]
Mr. And Mrs. Smith plan to roof the cabin on
2 consecutive days. Assuming that the chance of rain is
independent of the day, what is the probability that it
will rain both days?
A. 0. 04
B. 0. 08
C. 0. 16
D. 0. 20
E. 0. 40
From the given information provided, the probability that it will rain both days is 0.04 option A.
Since we are assuming that the chance of rain is independent of the day, we can use the multiplication rule of probability to find the probability that it will rain on both days.
Let's assume that the probability of rain on any given day is p. Then, the probability of no rain on that day is 1-p.
Therefore, the probability that it will rain on both days is:
P(rain on both days) = P(rain on day 1) × P(rain on day 2)
= p × p
= p²
Since the problem does not give us a specific value of p, we cannot determine the exact probability of rain on both days. However, we can use one of the answer choices to estimate the probability of rain on both days.
Looking at the answer choices, the only choice that is a perfect square is 0.04. Therefore, we can assume that p² = 0.04, which means that p = 0.2.
So, if the probability of rain on any given day is 0.2, then the probability of rain on both days is:
P(rain on both days) = p²
= 0.2²
= 0.04
Therefore, the answer is A. 0.04.
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