Answer:
b = -8
Step-by-step explanation:
To determine the value of b, we can use the factoring method. We can factor the expression
(5b + 16)(b + 32) as (b + 8)(5b + 16).
Setting each factor to 0, we get b + 8 = 0 and 5b + 16 = 0.
Solving for b in the first equation, we get b = -8.
Substituting this value into the second equation, we get 5(-8) + 16 = 0, which simplifies to -40 + 16 = 0.
This equation is true only when the value of b is -8. Therefore, the value of b is -8.
∫
1
/
√
1
+
(
ln
x
)
2
d
x
The integral ∫1/√1 +(lnx) 2dx is -x+2xln(x)+c
What is Integration?An integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data
The given integral is ∫1/√1 +(lnx) 2dx
= ∫1 +2(lnx)dx
Apply the sum rule
∫f(x)±g(x)=∫f(x)±∫g(x)
= ∫1dx +2∫(lnx)dx
=x+2(xln(x)-x)+c
=-x+2xln(x)+c
Hence, the integral ∫1/√1 +(lnx) 2dx is -x+2xln(x)+c
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Write the expression for each
phrase.
6 rows of 8
Find a line that passes through the points (-1,3) and (5,-1)
y2-y1/x2-x1
= (-1-3)/(5+1)
= -4/6
= -2/3
y = -(2/3)x + c
3 = (2/3) + c
c = -2/3 + 3
c = -2/3 + 9/3
c = -7/3
y = (2/3)x + (7/3)
A contour map is shown for a function f on the square R = [0, 8] [0, 8].
(a) Use the Midpoint Rule with m = n = 2 to estimate the value of ∫∫R f(x,y) dA. (Round your answer to the nearest integer.)
1360 ×
(b) Estimate the average value of f. (Round your answer to one decimal place.)
21.3 ×
The average value of f is 16.25.
What is contour map?A contour line of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value. It is a plane section of the three-dimensional graph of the function f parallel to the-plane.
Given that, a contour map is shown for a function f on the square R = [0, 8] [0, 8].
Since the given square is 8 by 8 and we are using m=n=2, then each sub-rectangle is 4 by 4. So, the area of the given rectangle is 64 and the area of each sub-rectangle is 16, that is, ΔA=16
Furthermore, the centers of the sub-rectangles are (2,2),(2,6),(6,2),(6,6).
a) The estimated value of [tex]\int\limits\int\limits_R f{x} \, dA[/tex]
=16⋅f(2,2)+16⋅f(2,6)+16⋅f(6,2)+16⋅f(6,6)
≈16⋅27+16⋅3+16⋅12+16⋅23
≈1040
b) The average value of f will be
[tex]f_{avg}=\frac{1}{A}.\int\limits\int\limits_R (x, y) \, dA[/tex]
≈ 1/64 ×1040
≈ 16.25
Therefore, the average value of f is 16.25.
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Find the difference between Mary's weight when she was 10 year's old and when she was born
The difference between Mary's weight when she was 10 year's old and when she was born will be;
⇒ 72 pounds
What is mean by Subtraction?Subtraction in mathematics means that is taking something away from a group or number of objects. When you subtract, what is left in the group becomes less.
Given that;
When Mary was born, she weighed 8 pounds.
And, When she was 10 years old, she weighed 10 times as much.
Now,
Since, When she was 10 years old, she weighed 10 times as much.
Hence, When she was 10 years old, she weighed = 10 × 8 pounds
= 80 pounds
And, When Mary was born, she weighed 8 pounds.
Thus, The difference between Mary's weight when she was 10 year's old and when she was born is,
⇒ 80 - 8 pounds
⇒ 72 pounds
So, The difference between Mary's weight when she was 10 year's old and when she was born is 72 pounds.
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The complete question is this,
When Mary was born, she weighed 8 pounds. When she was 10 years old, she weighed 10 times as much. How much more did Mary weigh when she was 10 years old than she was born?
The owner-manager of Good Guys Enterprises obtains utility from income (profit) and from having the firm behave in a socially conscious manner, such as making charitable contributions or civic expenditures. Can you set up the problem and derive the optimization conditions if the owner-manager wishes to obtain a specific level of utility at the lowest possible cost? Do these conditions differ from the utility maximizing conditions?
Answer:
Step-by-step explanation:
Certainly! To set up the problem, we can consider the owner-manager's utility as a function of two variables: income (profit) and charitable contributions or civic expenditures. Let's call these variables x and y, respectively.
The owner-manager utility is given by the function U(x,y), where x is the income (profit) and y is the charitable contributions or civic expenditures.
The cost of obtaining this utility is given by the function C(x,y), where x is the income (profit) and y is the charitable contributions or civic expenditures.
To find the lowest possible cost at which the owner-manager can obtain a specific level of utility, we can use the following optimization conditions:
The cost function C(x,y) should be minimized subject to the constraint U(x,y) = k, where k is the specific level of utility that the owner-manager wishes to achieve.
The first-order necessary conditions for a minimum are:
∂C/∂x = λ ∂U/∂x
∂C/∂y = λ ∂U/∂y
where λ is a Lagrange multiplier.
These conditions are known as the Karush-Kuhn-Tucker (KKT) conditions.
These optimization conditions differ from the utility-maximizing conditions in that we are minimizing the cost function instead of maximizing the utility function. In addition, we have the constraint U(x,y) = k, which means that we are trying to achieve a specific level of utility rather than trying to maximize it.
1) Use your own words to explain the following terms:
function,
domain, range vertical line test.
Function: A function is a mathematical relationship between two sets of values, where each element in the first set (called the domain) is related to exactly one element in the second set (called the range). A function can be represented by a formula, equation, or graph, and it describes how one set of values is related to another.
Domain: In a function, the domain is the set of all possible values that can be used as input for the function. It is the set of values that the function operates on.
Range: In a function, the range is the set of all possible values that can be produced as output by the function. It is the set of values that the function produces as a result of operating on the domain.
Vertical line test: The vertical line test is a method used to determine whether a graph represents a function.
Hope This Helps You!
Which equation is part of the same fact family as 5 x 4 =20?
Select the graph that correctly displays the solution of the system of inequalities. y > 2x – 4 y ≤ –x – 7
The solution to the system of given inequalities is (-7, 0).
What are inequalities?Inequalities are the mathematical expressions in which both sides are not equal. In inequality, unlike in equations, we compare two values. The equal sign in between is replaced by less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign.
The given system of inequalities are y> 2x-4 and y≤ -x-7.
The solution of the system of inequalities is the intersection region of all the solutions in the system.
x<y/2 +2 and x≤-y-7
The solution of the system of inequalities is (-7, 0)
Therefore, the solution to the system of given inequalities is (-7, 0).
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It cost the shoe factory $26.47 to make a pair of Purple Pedi running shoes. The mall sells them for $128.98. What is the percent mark-up?
If it cost the shoe factory $26.47 to make a pair of Purple Pedi running shoes and the mall sells them for $128.98 then 3.87 is the percent mark-up.
How much of a markup is that?The gross profit of a unit is determined by subtracting its cost to produce or acquire for resale from its sales price to determine the markup percentage. The markup percentage is computed as (12 - 8) / 8 and equals 50% if an item is priced at $12 but costs the manufacturer $8 to produce.
Some individuals will inform you that a 5% markup equates to a markup of 5% of the price your customer pays. In your illustration, this is stating that you add a specific amount to your cost, $62, to arrive at a price of $P for your customer, and the amount you added is 5% of $P. This means that your cost of $62 is 95% of the price of P.
($128.98 - $26.47)/ $26.47
= 3.87.
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y=x-4 and y=4x+2 graph solution of the systems
Answer:
(-2,-6)
Step-by-step explanation:
we can solve this system of equations with the substitution method because y is already by itself.
since they are both equal to y, we can set both equations equal to each other.
4x+2=x-4
subtract 2 from both sides
4x=x-6
subtract x from both sides
3x=-6
divide both sides by 3
x = -2
now that we know x, we can plug this value into one of the original equations and solve for y.
y=-2-4
y=-6
so on a graph, these two equations will intersect at the point (-2,-6)
1. Jason made 3 quarts of corn chowder. He divided it into serving sizes of 1 12 cups each. How many 1 12 -cup servings can be made from 3 quarts of chowder? A. 4 12 servings B. 8 servings C. 12 servings D. 18 servings
Answer:
B. 8 servings
Step-by-step explanation:
1 quart = 4 cups
3x4=12
3 quarts of clam chowder = 12 cups of chowder
1 1/2 cups each
12 divided by 1 1/2 = 12 divided by 3/2 = 12 divided by 2/3 = 8 servings
(20 points)which of these equations doesn’t fit with each other with what we’ve been learning about perpendicular lines explain why for full credit, equations in photo
Answer:
Line D
Step-by-step explanation:
You want to know which line is not parallel or perpendicular to the other given lines:
A: 2y -x = 2
B: 4y = -8x -4
C: y = -2x +4
D: 5y = 10x +15
Standard formWe can write each of these equations in standard form to make it easier to compare them. That form is ...
ax +by = c
where a > 0 and a, b, c are mutually prime.
The equations are ...
A: x -2y = -2
B: 2x +y = -1
C: 2x +y = 4
D: 2x -y = -3
Parallel linesEquations with the same coefficients of x and y will be parallel:
2x +y = -1 and 2x +y = 4 . . . . lines B and C
Perpendicular linesEquations with coefficients that have been swapped, with one negated, will be perpendicular:
x -2y = -2 and 2x +y = -1 . . . . lines A and B
Line A is perpendicular to parallel lines B and C, so line D is the odd one out.
Line D doesn't fit.
I need help with this!
Solve the following quadratic equation for all values of xx in simplest form.
4(x^2-5)-1=
4(x
2
−5)−1=
\,\,3
3
The quadratic equation 4(x² − 5) − 1 = 3 has its solutions to be are √6 and -√6
How to determine the solutions to the equation?From the question, we have the following parameters that can be used in our computation:
4(x 2−5)−1=3
Express properly
This gives
4(x² − 5) − 1 = 3
Subtract -1 from both sides of the equation
So, we have the following representation
4(x² − 5) = 4
Divide both sides by 4
This gives
x² − 5 = 1
Subtract -5 from both sides of the equation
So, we have the following representation
x² = 6
Take the square root of both sides
x = ±√6
Hence, the solution is ±√6
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Which graph represents a linear function?
The linear function is a straight line graph . Consequently, the third graph shows a linear function.
what is linear function ?When the independent variable (often 'x') varies by a constant amount and the dependent variable (typically expressed by 'y') changes by a constant amount, the function is said to be linear. To show the relationship between two quantities, a linear graph is a straight line graph. This graph is useful for showing a result as a series of simple straight lines. The phrase "linear" refers to a straight line and excludes the usage of curves, dots, bars, etc. Plot coordinates at positions where x and y are identical in order to draw the line y = x. More points can be plotted even though three are plenty.
given
A linear function has a graph that is a straight line. A nonlinear function, then, has a graph that is not a straight line.
Something like y equals x cubed or y equals e to the power of x is an illustration of this. Furthermore, if we look closely at our graph, we can see that the line is not vertical.
The graph of a linear function is a straight line. Consequently, the third graph shows a linear function.
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f(x)=3^x+1 and g(x) = 2x-4, find f(2) - g(-1)
Answer:
16
Step-by-step explanation:
3^x +1 = f(X)
2x-4 = g(x)
find f(2)-g(-1)
we substitute in function f(X) by value (2), and in function g(X) by value (-1)
1. 3^x +1 becomes 3^2+1 =9+1=10
2. 2(-1)-4= -2-4=-6
then f(2)-g(-1)
= 10-(-6) =10+6=16
Which values are possible rational roots of 9x3+14x2−x+18=0 according to the rational root theorem? Select each correct answer.
The possible roots of the cubic polynomial 9x³ + 14x² - x + 18 are 1, 2, and 3.
What is the rational root theorem?The rational root theorem explains how a polynomial's roots and coefficients are related. It explains the kind of rational roots that the polynomial might have in particular.
According to the rational root theorem, the possible roots of a polynomial of degree three and above are,
± (coefficient of the last term)/(coefficient of the first terms).
Given, A polynomial 9x³ + 14x² - x + 18 it is a cubic polynomial so it should have three roots they could be,
The factors of the constant term divided by the factors of the first terms,
So, 18 has factors ± (1, 2, 3, 6, 9, 18) and 9 has factors ± (1, 3, 9).
So, The possible roots are ± (1, 2, 3) and so on.
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I’ve been stuck on this for about a week, I know it might seem easy but I’m very bad at fractions and my teacher won’t help me
On solving the provided question we can say that - the increased value in fraction will be [tex]3\frac{1}{4}[/tex]
what is fraction?Any number of equal portions, or fractions, can be used to represent a whole. Fractions in standard English indicate how many units of a certain size there are. 8, 3/4. A whole includes fractions. The ratio of the numerator to the denominator is how numbers are expressed in mathematics. Each of these is an integer in simple fractions. In the numerator or denominator of a complex fraction is a fraction. True fractions have numerators that are less than their denominators. A fraction is a sum that constitutes a portion of a total. By breaking the entire up into smaller bits, you can evaluate it. Half of a full number or item, for instance, is represented as 12.
we have,
initial = [tex]4\frac{5}{6}[/tex]
increased by = [tex]4\frac{1}{4}[/tex]
son difference is = [tex]3\frac{1}{4}[/tex]
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eva invest $120 at a rate of 3% per year compound interest. calculate the total amount eva has after 2 years
Answer:
Eva has $127.31 after 2 years.
Step-by-step explanation:
Here is the formula for compound interest
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]
Where [tex]A[/tex] is the final amount, [tex]P[/tex] is the initial amount, [tex]r[/tex] is the interest rate as a decimal, [tex]n[/tex] is the number of times the interest applied per time period, and [tex]t[/tex] is the number of time periods.
We are given
[tex]P=120\\r=0.03\\n=1\\t=2[/tex]
Substitute our numbers into the equation
[tex]A=120(1+\frac{0.03}{1})^{1*2}[/tex]
[tex]A=120(1.03)^{2}[/tex]
[tex]A=120*1.03^{2}[/tex]
[tex]A=120*1.0609[/tex]
[tex]A=127.308[/tex]
13.A rectangle and a square have the same perimeter 120 cm. Find the side of the square. If the rectangle has a breadth 5cm less than that of the square. Find the breadth, length and area of the rectangle.
Answer:
Step-by-step explanation:
The population of North Carolina was approximately 9.5 million in 2010. The population has increased by about 1.3% per year. If the population continues to grow at this rate, in what year will North Carolina’s population reach 12 million?
The population of North Carolina will reach 12 million in the year 2028.
What is Population Growth?Population growth is defined as the increase in the number of people which is calculated using population growth rate.
Given that,
Population in 2010 = 9.5 million
Population is increasing to 1.3% per year.
Annual growth rate = 1.3% = 1.3/100 = 0.013
We have the population growth formula,
P = P₀ e^(r t)
Here, P₀ = 9.5 million = 9,500,000
and P = 12 million = 12,000,000
r = 0.013
We have to find t.
12,000,000 = 9,500,000 e^(0.013t)
120 / 95 = e^(0.013t)
Taking natural logarithm on both sides and also ㏑(e^m) = m ㏑ e, we get,
㏑(12/9.5) = 0.013t × ㏑e
[㏑(12/9.5) ÷ 0.013] = t [since ln e = 1]
t = 17.97 ≈ 18 years
The year is 2010 + 18 = 2028
Hence in the year 2028, the population will be 12 million.
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PS lies on the perpendicular bisector of QR. Select all the statements about the figure that must be true.
Of the given statements, only the option number {5} and {6} are true.
What is perpendicular bisector?The perpendicular bisectors of a triangle are lines passing through the midpoint of each side which are perpendicular to the given side. A triangle's three perpendicular bisectors meet at a point. known as the circumcenter, which is also the center of the triangle's circumcircle.
Given is a triangle as shown in the image.
Since PS is the perpendicular bisector of QR, we can write -
QS = SR
6n + 3 = 4n + 11
2n = 8
n = 4
QR = QS + SR
QR = 6n + 3 + 4n + 11
QR = 10n + 14
QR = 10 x 4 + 14
QR = 54
and
SR = 4n + 11
SR = 16 + 11
SR = 27
Therefore, of the given statements, only the option number {5} and {6} are true.
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Anna and Sharon both construct a triangle. Anna begins by drawing a segment with a length of 3 inches. Starting from one vertex, she draws another segment with a length of 5 inches so that the two segments have an included angle of 35°. Sharon constructs her triangle by drawing a segment with a length of 5 inches, measuring a 35° angle, and drawing a ray from one vertex. Then she measures and terminates the ray at 3 inches. Do Anna and Sharon create congruent triangles? Explain.
Answer:
Yes, Anna and Sharon both created triangles that are congruent. Their triangles are congruent because they have 2 corresponding, congruent side-lengths and an included angle that is congruent. Therefore, their triangles are congruent by SAS (Side-Angle-Side).
Consider the line whose equation is y=x+3.
Fill in the table below and then plot the line.
For the given line whose equation is [tex]y=x+3[/tex] :
x y (x,y)
0 3 (0,3)
1 4 (1,4)
2 5 (2,5)
3 6 (3,6)
What is equation?The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.
Define expression.Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation. An expression's structure is as follows: Expression: (Math Operator, Number/Variable, Math Operator)
A line in slope-intercept form, y = mx + b, where m is the slope of the line and b is the y-intercept .
Given the equation y = x + 3, the slope (m) is 1 and the y-intercept (b) is 3.
So, we can fill the table like this:
For the given line whose equation is [tex]y=x+3[/tex] :
x y (x,y)
0 3 (0,3)
1 4 (1,4)
2 5 (2,5)
3 6 (3,6)
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When Bashir commutes to work, the amount of time it takes him to arrive is normally distributed with a mean of 32 minutes and a standard deviation of 3 minutes. What percentage of his commutes will be shorter than 32 minutes, to the nearest 10th?
A percentage of his commute that would be shorter than 32 minutes, to the nearest 10th is equal to 50%.
How to determine the required percentage?Mathematically, the z-score of a given sample size or data set can be calculated by using this formula:
Z-score, z = (x - μ)/σ
Where:
σ represents the standard deviation.x represents the sample score.μ represents the mean score.Substituting the parameters into the z-score formula, we have;
Z-score, z = (32 - 32)/3
Z-score, z = 0/3
Z-score, z = 0
Since a commute of 32 minutes on a test corresponds to a z-score of 0, the percentage of the commute will be shorter than 32 minutes is given by:
P(x > 32) = P(z > 0)
P(z > 0) = 0.5
P(z > 0) = 50%.
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hello i need help with my math question one only pease and thank you
a) The after-tax cost of the vehicle Rachel is buying is $29,368.70.
b) The part of the after-tax cost that Rachel will finance after making the down payment is $25,118.70.
c) The total payments after 5 years are $30,659.60.
d) The total interest Rachel paid is $1,290.90.
What is the interest payment?The interest payment represents the difference between the total monthly payments and the financing part of the after-tax cost of the vehicle.
The total monthly payments ($26,409.60) are obtained by multiplying the monthly payments ($440.16) by the number of payment periods (60).
Value of vehicle = $25,990
Harmonized sales tax (HST) = 13%
After-tax cost of the vehicle = $29,368.70 ($25,990 x 1.13)
Down payment = $4,250
Financing part = $25,118.70 ($29,368.70 - $4,250)
Monthly payments = $440.16
Financing period = 60 months (5 years x 12)
Total monthly payments = $26,409.60 ($440.16 x 60)
The total amount paid for the car = $30,659.60 ($26,409.60 + $4,250)
Total interest = $1,290.90 ($26,409.60 - $25,118.70)
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How can you use this pattern to find 3°
Number pattern, sequence, finding the algebraic rule and using a ... To illustrate this, consider the following sequence of numbers {1, 3, 5, 7, 9, …}.
What is Number pattern?Number pattern is a pattern or sequence in a series of numbers. This pattern generally establishes a common relationship between all numbers.Here, we get the numbers in the pattern by skip counting by 5. Given are the steps to identify a number pattern.To solve the problems of number pattern, we need first to find the rule being followed in the pattern.
To find out the rule, we need to see the first few numbers in the series.Try to see the difference between consecutive numbers, it will help us understand the relationship between the numbers.Number pattern is a pattern or sequence in a series of numbers. This pattern generally establishes a common relationship between all numbers. For example: 0, 5, 10, 15, 20, 25, … Here, we get the numbers in the pattern by skip counting by 5.
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The complete question is -
How can you use this pattern to find 3° of the number in Series ?
The value of a second-hand car is £8,000.
Each year it loses 20% of its value.
Work out the value of the car after 5 years.
The value of the car will be £2,621 after 5 years.
What is exponential decay?Exponential decay describes the process of reducing an amount by a consistent percentage rate over a period of time.
Given that, The value of a second-hand car is £8,000. Each year it loses 20% of its value, we are asked to find its value after 5 years.
This is a situation of exponential decay.
Hence, using the exponential decay formula =
A = P(1-r)ⁿ
Where A is final value, P is principal value, r is rate of decrease and n is the number of years.
Here, we will find A, and we have,
P = £8,000.
r = 20% = 0.20
t = 5
Therefore,
A = 8000(1-0.20)⁵
A = 8000(0.80)⁵
A = 8000(0.32768)
A ≈ 2,621
Hence, the value of the car will be £2,621 after 5 years.
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A construction crew is lengthening a road. Let y represent the total length of the road (in miles). Let x represent the number of days the crew has worked.
Suppose that x and y are related by the equation y = 53+2x.
Answer the questions below.
Note that a change can be an increase or a decrease.
For an increase, use a positive number. For a decrease, use a negative number.
What is the change per day in the road's length?
miles
What was the road's length when the crew started working?
Answer: The equation y = 53+2x relates the variables x and y, where x represents the number of days the crew has worked and y represents the total length of the road.
The change per day in the road's length is given by the coefficient of x in the equation, which is 2. Thus, the change per day in the road's length is 2 miles.
The road's length when the crew started working is given by the constant term in the equation, which is 53. Thus, the road's length when the crew started working was 53 miles.
Step-by-step explanation: