We have to find the minimum number, k, of z in (a, b) such that g'(z) = 0. The function g is continuous on the interval [a, b] and is differentiable on (a, b). Suppose that g(x) = 0 for 4 distinct values of x in (a, b).
Let x1, x2, x3, and x4 be the four distinct values of x such that g(x) = 0.Now consider the following cases:Case 1: All four x1, x2, x3, x4 are local extrema of g(x).If this is the case, then g′(x1)=g′(x2)=g′(x3)=g′(x4)=0. Therefore, the minimum number, k, of z in (a, b) such that g′(z) = 0 is 4.Case 2:
There are less than four local extrema of g(x).In this case, by Rolle's Theorem, there exists at least one point z in (a, b) such that g′(z)=0. Since there are less than four local extrema of g(x), this point z is not equal to any of x1, x2, x3, and x4. Therefore, the minimum number, k, of z in (a, b) such that g′(z) = 0 is 1.In conclusion, the minimum number, k, of z in (a, b) such that g′(z) = 0 is either 1 or 4 depending on whether there are less than four local extrema of g(x) or all four x1, x2, x3, and x4 are local extrema of g(x).
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1. For one history test, Mary had to answer 40 questions, and Mary answered 38 of them correctly. What percent did Mary get on her History test? Round your answer to the nearest whole number if necessary.
Answer: 95%
Step-by-step explanation: 38 / 40 = 0.95
The population of a town grows at a rate proportional to the population present at time t. The initial population of 500 increases by 10% in 10 years. what will be the population in 60 years? (Round your answer to the nearest person.) ______ persons How fast is the population growing at t = 607(Round your answer to two decimal places.) ______persons/yr
When the initial population increases by 10% in 10 years, it is growing at a rate of approximately 78.04 persons per year at t = 60.
To find the population in 60 years, we need to use the formula:
P(t) = P0rt
P0 is the initial population
r is the rate of growth
t is the time in years.
So, given that the initial population of 500 increases by 10% in 10 years, we can find r as follows:
10% increase in 10 years means that the population has grown to (100% + 10%) = 110% of its original size in 10 years.
Therefore, we have:
P(10) = 500(1 + 0.10)
= 550
Now we can use these values to solve for:
r: 550 = 500er
⇒ er = 550/500
⇒ r = ln(550/500)/10
= 0.04879 (rounded to 5 decimal places)
Therefore, the population in 60 years is:
P(60) = 500e0.04879 × 60 ≈ 1599 (rounded to the nearest person)
The population is growing at a rate of:
P'(t) = rP(t),
so at t = 60, we have:
P'(60) = 0.04879 × 1599 ≈ 78.04 (rounded to two decimal places)
Therefore, the population is growing at a rate of approximately 78.04 persons per year at t = 60.
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Simplify 4 triangles to 16 squares
The expression 4 triangles to 16 squares when simplified is 1 triangle to 4 squares
How to simplify the expressionGiven that
4 triangles to 16 squares
When expressed as ratio, we have
Triangle : Square = 4 : 16
To simplify the ratio Triangle : Square = 4 : 16, we can divide both the numerator and denominator by their greatest common factor, which is 4.
So, we have
Triangle : Square = 4 : 16
Divide both sides by 4:
Triangle/4 : Square/4 = 1 : 4
So the simplified ratio is 1 : 4, which means for every one triangle, there are four squares.
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which part of this graph shows a non-linear relationship
Answer:
A.
Step-by-step explanation:
Factorise fully - 4x² - 16x
Answer: 4x(x - 4)
Step-by-step explanation:
4x² - 16x = 4x(x - 4)
Now we can see that the expression inside the parentheses can also be factored:
x - 4 = (x - 4)
So the fully factorized expression is:
4x² - 16x = 4x(x - 4) = 4x(x - 4)
Answer:
ㅤ
- 4x( x + 4 )
ㅤ
Step-by-step explanation:
ㅤ
[tex]\large{\pmb{\sf{ - 4x^{2} - 16x}}}[/tex]
ㅤ
[tex]\large{\underline{\underline{\sf{Taking \: Out \: {\green{4}} \: As \: Common:-}}}}[/tex]
ㅤ
[tex]\large{\pmb{\sf{\leadsto{- 4(x^{2} + 4x)}}}}[/tex]
ㅤ
[tex]\large{\underline{\underline{\sf{Taking \: Out \: {\green{x}} \: As \: Common:-}}}}[/tex]
ㅤ
[tex]\large{\purple{\boxed{\pmb{\sf{\leadsto{- 4x(x + 4)}}}}}}[/tex]
━━━━━━━━━━━━━━━━━━━━━━
[tex]\star \: {\large{\underline{\underline{\pink{\mathfrak{More:-}}}}}} \: \star[/tex]
ㅤ
[tex]\large{\dashrightarrow}[/tex] Two positive always makes positive sign when multiplied.
ㅤ
[tex]\large{\dashrightarrow}[/tex] Two negatives always makes positive sign when multiplied.
ㅤ
[tex]\large{\dashrightarrow}[/tex] A positive and a negative always makes negative sign when multiplied.
ㅤ
[tex]\large{\dashrightarrow}[/tex] The sum of two positives is always positive with a positive sign.
ㅤ
[tex]\large{\dashrightarrow}[/tex] The sum of two negatives is always positive with a negative sign.
ㅤ
[tex]\large{\dashrightarrow}[/tex] The sum of a positive and a negative is always negative with the sign of whose number is greater.
Find the missing side of each triangle round your answers to the nearest 10th
Please help it’s for tmr
Leo has a number of toy soldiers between 27 and 54. If you want to group them four by four, there are none left, seven by seven, 6 remain, five by five, 3 remain. How many toy soldiers are there?
The answer is 48 but I need step by step explanation
Hence, 28 toy soldiers are the correct answer.
In mathematics, how is a group defined?A group in mathematics is created by combining a set with a binary operation. For instance, a group is formed by a set of integers with an arithmetic operation and a group is also formed by a set of real numbers with a differential operator.
Let's refer to the quantity of toy soldiers as "x".
We are aware that x is within the range of 27 and 54 thanks to the problem.
x can be divided by 4 without any remainders.
The residual is 6 when x is divided by 7.
The leftover after dividing x by five is three.
These criteria allow us to construct an equation system and find x.
Firstly, we are aware that x can be divided by 4 without any residual. As a result, x needs to have a multiple of 4. We can phrase this as:
x = 4k, where k is some integer.
Secondly, we understand that the remaining is 6 when x is divided by 7. This can be stated as follows:
x ≡ 6 (mod 7)
This indicates that x is a multiple of 7 that is 6 more than. We can solve this problem by substituting x = 4k:
4k ≡ 6 (mod 7)
We can attempt several values of k until we discover one that makes sense for this equation in order to solve for k. We can enter k in to equation starting using k = 1, as follows:
4(1) ≡ 6 (mod 7)
4 ≡ 6 (mod 7)
It is not true; thus we need to attempt a next value for k. This procedure can be carried out repeatedly until the equation is satisfied for all values of k.
k = 2:
4(2) ≡ 6 (mod 7)
1 ≡ 6 (mod 7)
k = 3:
4(3) ≡ 6 (mod 7)
5 ≡ 6 (mod 7)
k = 4:
4(4) ≡ 6 (mod 7)
2 ≡ 6 (mod 7)
k = 5:
4(5) ≡ 6 (mod 7)
6 ≡ 6 (mod 7)
k = 6:
4(6) ≡ 6 (mod 7)
3 ≡ 6 (mod 7)
k = 7:
4(7) ≡ 6 (mod 7)
0 ≡ 6 (mod 7)
We have discovered that the equation 4k 6 (mod 7) is fulfilled when k = 7. Thus, we can change k = 7 to x = 4k to determine that:
x = 4(7) = 28
This indicates that there are 28 toy troops. Yet we also understand that the leftover is 3 when x is divided by 5. We don't need to take into account any other values of x because x = 28 satisfies this requirement.
28 toy soldiers are the correct response.
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Mia volunteers on the weekend at the Central Library. As a school project, she decides to record how many people visit the library, and where they go. On Saturday, 470 people went to The Youth Wing, 413 people went to Social Issues, and 350 went to Fiction and Literature. On Sunday, the library had 400 total visitors. Based on what Mia had recorded on Saturday, about how many people should be expected to go to The Youth Wing? Round your answer to the nearest whole number
Based on what Mia recorded, we calculate that 152 people are expected to go to the Youth wing on Sunday.
We solve this problem using simple arithmetic methods. According to the data collected by mia,
The number of people who went to the "Youth wing" = 470
The number of people who went to "Social issues" = 413
The number of people who went to "Fiction and Literature" = 350
So, the total number of visitors that the library had on Saturday was,
470+413+350 = 1233
Hence, the proportion of people who went to the "Youth wing", "Social issues" and "Fiction and Literature" are (470/1233), (413/1233), (and 350/1233) respectively.
So, on Sunday the expected number of people who may go to the "Youth wing" should be,
(470/1233)×400
= 152.47 ≈ 152
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I need the answer to this question(PLEASE IM BEGGING YOU)
Answer:
B & D
Step-by-step explanation:
For future reference, you can just a site called desmos. It has a graphing tool where you can just write the function and then check where the lines meet.
3. The population of bees has been increasing by 5% each year since 2010. There were 1,000 bees counted in 2010. a. Create an explicit formula that models the bee population for n years since 2010. (Hint: What function have we studied that can be represented by this situation?)
Answer: 1000 x 1.05^n
Step-by-step explanation:
1.05 is the increase experienced each year and n is the number of years.
Answer:
The population of bees is increasing by 5% each year since 2010. Let P(n) be the population of bees in the nth year since 2010.
The explicit formula for the bee population can be found using the formula for compound interest:
P(n) = P(0) * (1 + r)^n
where P(0) is the initial population in 2010, r is the annual growth rate, and n is the number of years since 2010.
Substituting the given values, we have:
P(n) = 1000 * (1 + 0.05)^n
Simplifying the expression, we get:
P(n) = 1000 * 1.05^n
Therefore, the explicit formula that models the bee population for n years since 2010 is P(n) = 1000 * 1.05^n.
The table shows the costs of different camping activities. Over the summer, Maura canoed 4 times and fished 3 times. Write and evaluate an expression that represents the total cost Maura spent canoeing
and fishing.
Answer:
Step-by-step explanation:
Here goes:
So, we want to write out an expression for this situation. I'm not exactly sure what you've been taught in class, but personally, I would start out with a let statement looking something like this:
Let x = the total cost Maura spent canoeing and fishing.
From there, we know you have 4 canoeing and 3 fishing trips. From here, we just plug in numbers for what we know. So, we get an equation that looks like this:
4(8) + 3(5) = x
Now, this is an equation, however, if you just had an expression like the question asked for, there wouldn't be anything to evaluate. Plus, it's always kind of satisfying to get a number answer. So, with a little bit of math, we get 32+15 = x, and a grand total of 47 = x.
Forgot what x was? Thank goodness you wrote a let statement. Look back up if you need the refresher.
Finally, if you need any other clarification, feel free to reach out with another question.
The village of Hampton has 436 families 238 of the families live within 1 mile of the village square use mental math to find how many families live farther than 1 mile from the square show your work
Answer: 198 families live farther than 1 mile from the square.
Step-by-step explanation:
We know that there are 238 families that live within 1 mile of the village square. To find the number of families that live farther than 1 mile from the square, we can subtract 238 from the total number of families:
436 - 238 = 198
Therefore, 198 families live farther than 1 mile from the square. We can do this subtraction mentally without needing a calculator.
a) Work out the minimum number of hikers who could have walked between 6 miles and 17 miles. b) Work out the maximum number of hikers who could have walked between 6 miles and 17 miles. < Back to task Distance, a (miles) 0≤ x<5 5 ≤ x < 10 10 ≤ a < 15 15 ≤ x < 20 20 ≤ w Scroll down Watch video Frequency 3 2 9 8 4 Answer
9 hikers are the bare minimum that might have covered the range of 6 to 17 miles because that distance falls inside the typical interval of 10 x 15 miles.
What is meant by minimum and maximum value?Rearrange the function using fundamental algebraic concepts to determine the value of x when the derivative equals 0.
This response gives the x-coordinate of the function's vertex, which is where the maximum or minimum will occur.
To determine the minimum or maximum, rewrite the solution into the original function.
The greatest and smallest values of a function, either within a specific range (the local or relative extrema) or throughout the entire domain, are collectively referred to as extrema (PL: extrema) in mathematical analysis.
b) the maximum number of hikers who could have walked between 6 miles and 17 miles is 19.
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we could, in principle, represent a polynomial as a list. for instance, we could write as [ 1,2,-3,0,2 ] where the ith index corresponds to . if we wrote a polynomial this way, we would also like an easy way to evaluate that polynomial for a specified value of ; i.e., for , compose a function polyeval( coefs,x ) which accepts a list of polynomial coefficients from lowest to highest order (as above) and a value x at which to evaluate the polynomial, and returns a float corresponding to the value of the polynomial evaluated at x.
The list above represents a polynomial of degree 4.The following polyeval( coefs,x ) function returns the value of the polynomial evaluated at x:1.
HTML representation of the polyeval( coefs,x ) function:```def polyeval(coefs, x): poly_sum = 0 for i in range(len(coefs)): poly_sum += coefs[i] * x**i return poly_sum```Explanation:A polynomial can be represented in the form of a list. In Python, the representation of a polynomial as a list [1, 2, -3, 0, 2] means that the ith index corresponds to the coefficient of the term of degree i. Therefore, the list above represents a polynomial of degree 4.The following polyeval( coefs,x ) function returns the value of the polynomial evaluated at x:1. The function accepts two parameters: a list of polynomial coefficients from lowest to highest order and a value x at which to evaluate the polynomial.2. The function returns a float corresponding to the value of the polynomial evaluated at x.
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a commercial kitchen uses 3/4 of a cup of milk every 4/6 of a minute. how many cups of milk are used per minute answer key
The amount of milk used per minute in a commercial kitchen that uses 3/4 of a cup of milk every 4/6 of a minute is 1/2 cup of milk.
How many cups of milk are used per minute?In a recipe or cooking, fractions are frequently used. We can use them to measure ingredients such as sugar, butter, milk, and other items. The numerator of the fraction refers to the number of parts that are utilized. The denominator, on the other hand, refers to the whole.
The fraction 3/4 can be defined in the following ways: 3 parts out of 4 parts,75 parts out of 100 parts,15 parts out of 20 parts,The fraction 4/6 can be reduced as:4/6 = (4 ÷ 2)/(6 ÷ 2) = 2/3Thus, the fraction 4/6 represents 2/3 or two parts out of three parts.
We can use proportions to figure out how many cups of milk are used per minute. To do that, we need to convert the given quantity of milk into a fraction that represents the amount of milk used per minute
The kitchen uses 3/4 of a cup of milk every 4/6 of a minute.=> The fraction that represents the amount of milk used per minute = [3/4 ÷ 4/6]=> Multiplying the numerator and denominator of the above fraction by 6, we get:[3/4 ÷ 4/6] = [3/4 × 6 ÷ 4/6 × 6] = [18/24 ÷ 24/24] = 18/24= 3/4 (Reduced Form)Therefore, 3/4 of a cup of milk is used per 4/6 of a minute, or 1/2 cup of milk per minute, if we simplify it further.
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answer quickly please
Answer:
m = -3/4
Step-by-step explanation:
Slope = rise/run or (y2 - y1) / (x2 - x1)
Pick 2 points (0, -2) (-3,2)
We see the y increase by 4 and the x decrease by 3, so the slope is
m = -3/4
See the image posted
The probability that both bulbs are red is 0.126 and The probability that the first bulb selected is red and the second yellow is 0.113
What is Probability?Probability means the possible outcome occur when an event take place.
(a) The probability that both bulbs are red ,
= 11/30 * 10/29
= 11/87
= 0.126
So, The probability that both bulbs are red is 0.126
(b) The probability that the first bulb selected is red and the second yellow,
= 11/30 * 9/29
= 33/290
= 0.113
So, The probability that the first bulb selected is red and the second yellow is 0.113
(c) The probability that the first bulb selected is yellow and the second red,
= 9/30 * 11/29
= 33/290
= 0.113
So, The probability that the first bulb selected is yellow and the second red is 0.113
(d) The probability that one bulb is red and the other yellow,
= 33/290 + 33/290 ( Add (b) and (c) )
= 33/145
= 0.227
And, The probability that one bulb is red and the other yellow is 0.227 .
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How can you describe the variability in the number of hours of sleep? Explain.
Hi can someone help me pls
Answer:
F(x) decreases by 6
Step-by-step explanation:
The two points are (-4,10) and (-1,4) and the line is going down as the x value increases. therefore subtract 4 from 10 to get a decrease of -6.
A light bulb manufacturer claims its light bulbs will last 500 hours on average. The lifetime of a light bulb is assumed to follow an exponential distribution. (15 points) a. What is the probability that the light bulb will have to be replaced within 500 hours? s. RSS THE b. What is the probability that the light bulb will last more than 1,000 hours? c. What is the probability that the light bulb will last between 200 and 800 hours?
a.There is a 63.21% chance that the light bulb will have to be replaced within 500 hours.
The probability that the light bulb will have to be replaced within 500 hours can be calculated by finding the area under the exponential probability density function (PDF) from 0 to 500. Using the formula for the exponential PDF with a mean of 500, we get:
P(X ≤ 500) = 1 - e^(-500/500) ≈ 0.6321
Therefore, there is a 63.21% chance that the light bulb will have to be replaced within 500 hours.
b. There is a 39.35% chance that the light bulb will last between 200 and 800 hours.
The probability that the light bulb will last more than 1,000 hours can be calculated by finding the area under the exponential PDF from 1000 to infinity. Using the same formula, we get:
P(X > 1000) = e^(-1000/500) ≈ 0.1353
Therefore, there is a 13.53% chance that the light bulb will last more than 1,000 hours.
c. The probability that the light bulb will last between 200 and 800 hours is0.3935.
It can be calculated by finding the area under the exponential PDF from 200 to 800. Again, using the same formula, we get:
P(200 < X < 800) = e^(-200/500) - e^(-800/500) ≈ 0.3935
Therefore, there is a 39.35% chance that the light bulb will last between 200 and 800 hours.
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a cup of hot coffee is placed outside where the temperature is 0, assume the coffee cools to approach the outside temperature according to an exponential decay model, if the continuous rate of cooling is determined to be 2 percent per minute and the current temperature of the coffee is 54.8 celsius how many minutes will the coffee cool to 44.9 Celsius
It will take approximately 27.7 minutes for the coffee to cool from 54.8°C to 44.9°C when following exponential decay model.
What is exponential decay?A quantity declines over time proportionate to its existing value through a process known as exponential decay. An exponential function of the form f(t) = ab raised to t, where an is the beginning value, b is the decay factor (a number between 0 and 1), and t represents time, mathematically describes this.
Several real-world circumstances, like population increase, radioactive decay, and the loss of electrical charge in a capacitor, exhibit exponential decay.
Given that the situation follows a exponential decay model.
The exponential decay is given as:
[tex]T(t) = T0 * e^{(-rt)}[/tex]
Substituting the values T0 = 54.8, r = 0.02, and T(t) = 44.9.
[tex]44.9 = 54.8 * e^{(-0.02t)}\\0..8208 = e^{(-0.02t)}\\ln(0.8208) = -0.02t\\t = ln(0.8208)/(-0.02) = 27.7 minutes[/tex]
Hence, it will take approximately 27.7 minutes for the coffee to cool from 54.8°C to 44.9°C.
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russell is doing some research before buying his first house. he is looking at two different areas of the city, and he wants to know if there is a significant difference between the mean prices of homes in the two areas. for the 40 homes he samples in the first area, the mean home price is $183,100. public records indicate that home prices in the first area have a population standard deviation of $21,845. for the 33 homes he samples in the second area, the mean home price is $161,500. again, public records show that home prices in the second area have a population standard deviation of $24,820. let population 1 be homes in the first area and population 2 be homes in the second area. construct a 95% confidence interval for the true difference between the mean home prices in the two areas. round the endpoints of the interval to the nearest whole number, if necessary.
Confidence Interval: The confidence interval for the true difference between the mean home prices in the two areas at a 95% confidence level is given as [tex]16,322 < \mu_1 - \mu_2 < 31,338.[/tex]
Given: Population 1: [tex]n_1 = 40, \mu_1 = 183,100, \sigma_1 = 21,845[/tex]
Population 2: [tex]n_2 = 33, \mu_2 = 161,500, \sigma2 = 24,820[/tex]
To construct a 95% confidence interval for the true difference between the mean home prices in the two areas, we need to calculate the sample mean difference between the two populations, as well as the standard error.
The sample mean difference is given by: [tex]\bar{x}_1 - \bar{x}_2 = 183,100 - 161,500 = 21,600[/tex]
The standard error is given by: [tex]s = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \\s= \sqrt{\frac{21,845^2}{40} + \frac{24,820^2}{33}} = 8,205[/tex]
Using the standard normal distribution table, with a confidence level of 95%, we obtain a z-score of ±1.96. Using these values, the 95% confidence interval for the true difference between the mean home prices in the two areas is given by: 21,600 - (1.96 * 8,205) < μ1 - μ2 < 21,600 + (1.96 * 8,205)
Solving for the inequality, we get 16,322 < μ1 - μ2 < 31,338
Therefore, the confidence interval for the true difference between the mean home prices in the two areas at a 95% confidence level is given as $16,322 < μ1 - μ2 < $31,338.
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what is the target domain for a poisson distribution?
The target domain for a Poisson distribution is given the term (0, inf) which can be seen correct in option B.
A Poisson distribution's target domain is (0, inf). This means that the Poisson distribution can only be specified for non-negative integer values of the random variable it is modelling.
The Poisson distribution is a discrete probability function, which indicates that the variable may only take particular values from a finite list of integers. A Poisson distribution estimates how many times an event will occur in "x" amount of time. In other words, it is the probability distribution resulting from the Poisson experiment.
A Poisson experiment is a statistical experiment that categorises the experiment as either successful or unsuccessful. A limiting process of the binomial distribution is the Poisson distribution.
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Complete question:
What is the target domain for a Poisson distribution?
1) (-inf, inf)
2) (0, inf)
3) (-inf, 0]
4) [0, inf)
Solve the system of equations shown below using graphing and substitution. y=2x+3 and y=15-x
Answer: -17x+3
Step-by-step explanation:
y=2x+3 and y=15-x
15x-2x+3
-17x+3
you can try this
find the product. Write your answer in scientific notation (5 x 10^-7) x (3 x 10^6)
Write 0. 0166 correct to two significant figures.
0.0166 correct to two significant figures is 0.016.
To write 0.0166 to two significant figures, we need to look at the first two significant digits of the number, which are 1 and 6.
Since the digit after 6 is less than 5, we round down the last significant digit (6) to get:
0.016
Therefore, 0.0166 correct to two significant figures is 0.016.
Significant figures (also known as the significant digits, precision or resolution) of a number in positional notation are digits in the number that are reliable and necessary to indicate the quantity of something.
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if the circumference of the moon is 6783 miles what is its diameter in miles
Answer:
C = 21,309.4
Step-by-step explanation:
Diameter of moon is miles is,
d = 2159.8 miles
We have,
The circumference of the moon is, 6783 miles
Since, We know that,
the circumference of circle is,
C = 2πr
Substitute given values,
6783 miles = 2 × 3.14 × r
6783 = 6.28 × r
r = 6783 / 6.28
r = 1079.9 miles
Therefore, Diameter of moon is miles is,
d = 2 x r
d = 2 x 1079.9
d = 2159.8 miles
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What geometric shapes can you draw that have exactly one pair of parallel sides? Use pencil and paper. Sketch examples for as many different types of shapes as you can. Use appropriate marks to show the pairs of parallel sides.
A. regular pentagon
B. square
C. Trapezoid
D. parallelogram
Step 1 of 3
a.
About 180,000 terawatts of solar power reaches Earth’s surface.
Out of which about 0.06% is used by plants for photosynthesis.
Thus 108 terawatts of solar power is used by plants for photosynthesis.
Of this energy, about ends up stored in plant matter
Thus 1.08 terawatts of energy get stored in plant matter.
Consider the following facts
Therefore,
Therefore joules of energy get stored in plant matter each second.
The amount of energy stored in plant matter each second is 1.08 × 10^12 Joules.
Step 1: Calculation of joules of energy stored in plant matter each secondGiven that, 1.08 terawatts of energy gets stored in plant matter for photosynthesis in a second.Therefore, 1.08 × 1012 watts of energy gets stored in plant matter each second. Also, the energy stored in plants matter in Joules = Watts × seconds (Joule is the unit of energy)1.08 × 1012 watts of energy stored in plant matter in a second.1 watt-second = 1 JouleEnergy stored in plant matter in a second = 1.08 × 1012 watts × 1 second = 1.08 × 1012 Joules Answer: The amount of energy stored in plant matter each second is 1.08 × 10^12 Joules.
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From the roof a house 10 m. high, a man observes two cars on the ground, both due west the same line at angles of depression of 45° and 30° .How far apart are the two cars? Find it.
Step-by-step explanation:
hey, you just changed the angles in the question.
this following answer was about the angles of depression of 15° and 30°.
you cannot change the problem, when the answers are already given for the original problem.
so, I will add a copy with the adapted numbers for 45° and 30° after my original answer.
this creates 2 right-angled triangles.
the right angle is in both cases the angle where house meets the ground.
they also share one leg : the height of the house (10 m).
the second legs are the ground distances of the cars from the house.
the 2 Hypotenuses are the line of sight from the roof to the corresponding car.
remember, the sum of all angles in a triangle is always 180°.
again, we know one angle : the 90° angle.
but we also know a second angle based on the angles of depression (the "downward looking angles").
the triangle internal angle at the rooftop is the complementary angle (the difference to 90°) of the angle of depression.
so, this is 90-15 = 75° and 90-30 = 60°.
the angles at the cars on the ground are then
angle car 1 = 180 - 90 - 75 = 15°
angle car 2 = 180 - 90 - 60 = 30°
now, remember the trigonometric triangle inscribed in a circle.
imagine that the vertex at the car is the center of the corresponding circle around the trigonometric triangle.
the height of the house is then sine of the angle at the car multiplied by the Hypotenuse (= the line is sight from the rooftop to the car), which is the angle creating radius of the circle.
and the ground distance is the cosine of that same angle multiplied by the Hypotenuse.
so, we need to get the ratio of the height of the house / sin(car angle) to get the length of the Hypotenuse (line of sight). with that we can then calculate the ground distance as cosine of the angle multiplied by the same Hypotenuse.
for car 1 we have
10m/sin(15) = 38.63703305... m line of sight
that means ground distance of car 1 is
cos(15)×38.63703305... = 37.32050808... m
for car 2 we have
10m/sin(30) = 20 m line of sight
that means ground distance of car 2 is
cos(30)×20 = 17.32050808... m
since both cars are driving on the same side of the house in the same direction, the distance between both cars is purely the difference between their distances from the house :
37.32050808... - 17.32050808... = 20 m
the cars are 20 m apart.
and now for the angles of depression of 45° and 30° :
the triangle internal angle at the rooftop is the complementary angle (the difference to 90°) of the angle of depression.
so, this is 90-45 = 45° and 90-30 = 60°.
the angles at the cars on the ground are then
angle car 1 = 180 - 90 - 45 = 45°
angle car 2 = 180 - 90 - 60 = 30°
now, remember the trigonometric triangle inscribed in a circle.
imagine that the vertex at the car is the center of the corresponding circle around the trigonometric triangle.
the height of the house is then sine of the angle at the car multiplied by the Hypotenuse (= the line is sight from the rooftop to the car), which is the angle creating radius of the circle.
and the ground distance is the cosine of that same angle multiplied by the Hypotenuse.
so, we need to get the ratio of the height of the house / sin(car angle) to get the length of the Hypotenuse (line of sight). with that we can then calculate the ground distance as cosine of the angle multiplied by the same Hypotenuse.
for car 1 we have
10m/sin(45) = 14.14213562... m line of sight
that means ground distance of car 1 is
cos(45)×14.14213562... = 10 m
logically, as for 45° sine and cosine are equal.
for car 2 we have
10m/sin(30) = 20 m line of sight
that means ground distance of car 2 is
cos(30)×20 = 17.32050808... m
since both cars are driving on the same side of the house in the same direction, the distance between both cars is purely the difference between their distances from the house :
17.32050808... - 10 = 7.32050808... m
≈ 7.32 m
the cars are about 7.32 m apart.