a 6 ft tall man walks at the rate of 5 ft/sec toward a street light that is 16 ft above the ground. a. at what rate is the length of his shadow changing when he is 10 ft from the base of the light? b. at what rate is the tip of his shadow moving?

The description of the cosine function is [tex]\mathrm{f(x) = -10 \cos( \frac{2\pi}{3}) x + 10}[/tex].

A cosine function's general form is given by:

[tex]\mathrm{f(x) = A \times cos(Bx + C)) + D}[/tex]

where:

A stand for amplitude, or the ratio of the maximum to the minimum value.

B = 2π / period

C stands for horizontal phase shift.

(Midpoint between the Maximum and Minimum Values) D = Vertical Shift

We can determine the parameters for the specified function given the information given:

Period = 3 (This means B = 2π / 3)

Maximum value = 20

This means the amplitude A is half of this, so A = 20 / 2 = 10

Minimum value = 0

This means the vertical shift D is half of the difference between the maximum and minimum.

So, D = (20 - 0) / 2

= 10

The amplitude is negative because the function is a mirror of its parent function over the x-axis, hence A = -10.

Putting it all together, the function is:

[tex]\mathrm{f(x) = -10 \cos( \frac{2\pi}{3}) x + 10}[/tex]

Hence the final expression after the transformation is [tex]\mathrm{f(x) = -10 \cos( \frac{2\pi}{3}) x + 10}[/tex].

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Answer:

Step-by-step explanation:

y=32x+14

I got this equation because the initial height is the starting so that would be the y-int which comes after the slope, the slope is 32 because it says that the plant has grown that amount of centimeters each day, and each day should refer to the rate meaning slope. That would be the equation that you would use to find the plant's height after a given number of days,

For example..

y=32(5) + 14

I plugged in a 5 for x meaning 5 days. so you'd multiply 32 by 5 which is 160 and then you'd add 14 and you'd get 174. Meaning that after 5 days the plant has grown 174 cm...

(hopefully that helps a bit..if not then I'm sorry)

on solving the provided question, we can say that - 5/13 measure shown in the triangle are equivalent to sin [tex]\alpha[/tex] and cos [tex]\beta[/tex].

Trigonometry uses the first six fundamental trigonometric functions. Trigonometric ratios are these functions. The six fundamental trigonometric functions are the sine function, cosine function, secant function, co-secant function, tangent function, and co-tangent function. A right-angled triangle's side ratio corresponds to the trigonometric functions and identities. In order to determine the sine, cosine, tangent, secant, and cotangent values, trigonometric formulae are applied to the perpendicular side, hypotenuse, and base of a right triangle.

as cos x = base/ hypotenuse

and, sin x = perpendicular/ hypotenuse

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Option- D is correct that is the probability that at most 27 heads occur is 0.99999994.

Given that,

A fair coin is tossed 29 times.

We have to find what is the probability that at most 27 heads occur.

We know that,

A fair coin is tossed 29 times.

n=29

Probability of heads p=1/2

q= 1-p

q=1-1/2=1/2

We get

P (X=x) = ⁿCₓ qⁿ⁻ˣ pˣ

Now, the probability that at most 27 heads occurs is

P(X<27)

=1-[P(X=28)+ P(X=29)]

=1-[²⁹C₂₈(1/2)²⁹⁻²⁸(1/2)²⁸- ²⁹C₂₉(1/2)²⁹⁻²⁹(1/2)²⁹]

=1-[²⁹C₂₈(1/2)²⁹- ²⁹C₂₉(1/2)²⁹]

=1-[28+1](1/2)²⁹

=1-29×0.00000000186

=0.999999945

Therefore, Option- D is correct that is the probability that at most 27 heads occur is 0.99999994.

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Answer:

To test the hypothesis that most students in Year 9 sleep at least 10 hours on a school night, a questionnaire could be designed with the following questions:

1) How many hours of sleep do you typically get on a school night?

2) On a scale of 1 to 10, with 1 being "not at all" and 10 being "extremely", how well-rested do you feel when you wake up in the morning?

3) Do you have any difficulties falling asleep at night?

4) Do you have any daytime sleepiness or fatigue?

5) Do you regularly engage in activities that interfere with your sleep, such as using electronic devices before bed or consuming caffeine late in the day?

The questionnaire should also include space for respondents to provide their age and grade level, as well as any additional comments or observations they may have about their sleep habits. By collecting and analyzing this data, it should be possible to determine whether the hypothesis is supported or not.

Caitlyn would need to know the population size of Montana in order to predict how many babies will be born this year in the state.

In order to predict how many babies will be born this year in Montana, Caitlyn needs to have access to the population size of Montana. The total fertility rate for Montana is 1.73, which is the average number of children a woman can expect to have throughout her lifetime. Knowing the population size of Montana will allow Caitlyn to have a better understanding of how many babies are likely to be born this year, as this number is directly correlated to the total fertility rate. By having access to both the population size and the total fertility rate, Caitlyn will be able to make a more informed prediction about the number of babies that will be born this year in Montana.

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The numbers x and y values are the integer x = 29,288 and y = 713.

What is a linear equation?

Ax+By=C represents a two-variable linear equation in its standard form.

A linear equation with one variable has the conventional form Ax + B = 0. x is a variable, A is a coefficient, and B is constant in this situation.

An equation is used to show the relationship between variables, and numbers.

Let's call the two numbers x and y, where x is the first number and y is the second number. We are given that x + y = 30,000 and that 0.07x + 0.1y = 2,400.

Using substitution, we can solve for one variable in terms of the other and then substitute this expression into the other equation.
From the first equation, we can solve for y: y = 30,000 - x
Substituting this expression into the second equation, we get:
0.07x + 0.1(30,000 - x) = 2,400
0.07x + 3,000 - 0.1x = 2,400
Combining like terms, we get:
-0.03x = -600
Dividing both sides by -0.03, we get:
x = 20,000
Substituting this value back into the first equation, we can solve for y:
y = 30,000 - x
y = 30,000 - 20,000
y = 10,000
Therefore, the two numbers are x = 20,000 and y = 10,000.
Alternatively, we could have used elimination to solve the system. We can do this by adding the two equations together:
x + y = 30,000
0.07x + 0.1y = 2,400
0.07x + 0.1y + x + y = 30,000 + 2,400
0.17x + 1.1y = 32,400
Dividing both sides by 0.17, we get:
x + 6.47y = 190,588
Multiplying both sides of the first equation by 6.47, we get:
6.47x + 6.47y = 194,490
Subtracting the second equation from the first, we get:
5.47y = 3,902
Dividing both sides by 5.47, we get:
y = 712.5
Substituting this value back into the first equation, we can solve for x:
x = 30,000 - y
x = 30,000 - 712.5
x = 29,287.5

Since x and y must be positive integers, we can round these values to the nearest integer to get x = 29,288 and y = 713. These values are very close to the actual solution, which is x = 20,000 and y = 10,000. This is because we introduced some error when we divided both sides of the equation 0.17x + 1.1y = 32,400 by 0.17.
Hence, the numbers x and y values are the integer x = 29,288 and y = 713.

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The value for x is 25.5 units.

What are the Basic Trigonometry Formulas?

Basic trigonometry formulas are used to find the relationship of trig ratios and the ratio of the corresponding sides of a right-angled triangle. There are basic 6 trigonometric ratios used in trigonometry, also called trigonometric functions- sine, cosine, secant, co-secant, tangent, and co-tangent, written as sin, cos, sec, csc, tan, cot in short.

We have,

Perpendicular side = 18

Hypotenuse side = x

θ = 90/2 = 45°

sinθ = Perpendicular side/Hypotenuse side

sinθ = 18 / x

sin(45°) = 18 / x

1/√2 = 18 / x

x = 18 *√2

x = 25.4558

x = 25.5

Hence, the value for x is 25.5 units.

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To find the percent of bags of tortilla chips that were fat free, we need to divide the number of fat-free bags by the total number of bags and multiply the result by 100%.

The percentage of bags that were fat-free is:

(2173 fat-free bags / 3481 total bags) * 100% = 62.72%

Rounded to the nearest percent, the percentage of bags that were fat-free is 63%.

Answer: About 62%

Step-by-step explanation: About 62% of the bags were fat-free. I hope this helps ;)

The two lines on the system aren't parallel, thus, the system has only one solution.

Here we have the following system of equations:

3x - 10y = -7

-17x - 10y = -16

Now, notice that both equations are linear equations, and the equations are not parallel (the coefficients on the left side are different), which means that the two linear equations will intercept at some point (and at only one point)

That is enough to conclude that the system of equations will have only one solution.

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Answer:

The ratio of x-values to y-values in the table is 2:3

Step-by-step explanation:

8 / 12 = 2/3

10 / 15 = 2/3

12 / 18 = 2/3

Each has a ratio of 2 to 3.

The expression for the sum is 2n + 1 and the sum must be an odd number because it will leave 1 as remainder when divided by 2.

The Euclid division lemma states that when two positive integers a and b are divided then their are other two integers q and r that represent quotient and remainder. The equation for the lemma can be written as a = bq + r where r < b.

(a) The given consecutive integers are n and n + 1 respectively.

The expression for their sum can be written as,

n + n + 1

⇒ 2n + 1

(b) The general expression for the sum is 2n + 1.

Now, as per the Euclid's division lemma, the number when divide by 2 will leave remainder as 1.

Which implies the number will be an odd number.

Hence, the expression for the sum is 2n + 1 and it will be an odd number.

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Answer:

u=14, v=2

Step-by-step explanation:

Here's my attempt:

Proving triangle congruency:

We need to prove that the two triangles are congruent.

First, we need to see which sides/angles correspond (basically how the two triangles overlap each other).

Well, we can see that m∠F = 90° and that m∠D = 90°, so we know that ∠F and ∠D are congruent, and so they correspond.

So, if you drag ΔHFG over to ΔCDE, then sides DC and FH would be right on top of each other. (try drawing this to make it clearer)

Solving for v:

So if sides DC and FH are congruent, then their values are also equal.

DC = 17v and FH = v+32

So if DC=FH, then 17v=v+32

Then, we simplify this:

17v=v+32     [original equation]

16v=32     [subtract v from both sides]

v=2     [divide both sides by 16 to isolate v]

So, we have our value for v: v=2

Solving for u:

Now, it's time to find u:  (also, it doesn't matter if you found v first or u first in this problem)

When we overlapped ΔHFG and ΔCDE, we saw that EC = GH

So, using the same method as we used to find v, we can say that

EC=7u and GH = 10u-42

If EC=GH, then 7u=10u-42

Then, we simplify:

7u=10u-42     [original equation]
-3u=-42     [subtract 10u from both sides to get u on just one side]
u=14     [subtract both sides by -3 to isolate u]

Finally, we have our value for u: u=14

But we're not done yet: let's check our work:

Firstly, let's plug our answer for v into our equation for v:

17v=v+32
17(2)=(2)+32
34=34     It works!

Next, we'll plug in our answer for u into our equation for u:

7u=10u-42

7(14)=10(14)-42

98=140-42

98=98     It works!

I hope you read through my explanation and that it helped! :D

Answer:

The slope is -9/8.

Step-by-step explanation:

[tex]m = \frac{13 - 4}{2 - 10} = \frac{9}{ - 8} = - \frac{9}{8} [/tex]

The diameter of the spherical tank that holds 9000ft³ of water is 25.81 ft.

A diameter is a line that divides a circle into two equal parts.

To calculate the diameter of the tank, we use the formula below.

Formula:

Making d the subject of the equation,

Where:

From the question,

Given:

Substitute these values into equation 2

Hence, the diameter of the tank is 25.81 ft.

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The values of P and a that will make f(x)=Pa^x to be an exponential growth function is; A P = 1²; a = 2

The general formula for exponential growth function is;

y = a(b)ˣ

When a is a positive value and 'b' is the base greater than 1,

In order to represent exponential growth, a must be a positive number larger than 1.  If a is a fraction less than 1, we are basically taking a percent (less than 100) of the number and this will make it smaller, which would the become a decay function.  If a is a negative, the values would alternate between positive and negative.

The only option where a is a positive number larger than 1 is P = 1²; a = 2

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The correct answer is 2x = 14; x = 7

Therefore, The correct answer is 2x = 14; x = 7

Total bill that should be paid by Olive is $46.61. Olive's calculation is incorrect because she is applying the discount and tax to the original price of the clothes rather than to the discounted price.

A total bill is the final amount of money that is due for payment, usually for goods or services that have been provided. It is the sum of all charges, fees, taxes, and other expenses that have been incurred. The total bill may be presented as a single amount or as a breakdown of the various charges that make up the total.

The correct expression to calculate the total cost of the clothes would be:

Total cost = (Original price * (1 - Discount rate)) * (1 + Tax rate)

In this case, the original price is $54.50, the discount rate is 20%, and the tax rate is 7.5%. Plugging these values into the above expression gives us:

Total cost = (54.50 * (1 - 0.20)) * (1 + 0.075)

Total cost = (54.50 * 0.80) * 1.075

Total cost = $43.60 * 1.075

Total cost = $46.61

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Answer: 18in

Step-by-step explanation:

0.50x 3 = 1.50

50 divided by 5 is 10

1in every 10 cent

Answer:

a = - 3

Step-by-step explanation:

r² - 6r + 9

consider the factors of the constant term (+ 9) which sum to give the coefficient of the r- term (- 6)

the factors are - 3 and - 3 , since

- 3 × - 3 = + 9 and - 3 - 3 = - 6 , then

r² - 6r + 9

= (r - 3)(r - 3)

= (r - 3)² ← in the form (r + a)²

with a = - 3

A & D are complex conjugates because the sum of A and D is a real number. Also, A & D are complex conjugate because the product of A and D is a real number.

Options A and D are the correct answer

The complex conjugate of a complex number is the number obtained by changing the sign of its imaginary part. The complex conjugate is written with a bar over the complex number, like this:

z = a + bi

z* = a - bi

where a and b are real numbers and i is the imaginary unit, which is defined as the square root of -1

For example, the complex conjugate of the complex number 3 + 4i is 3 - 4i.

The complex conjugate has several important properties. For example, the product of a complex number and its complex conjugate is always a real number:

(a + bi)(a - bi) = a²+ b²

Also, the sum of a complex number and its complex conjugate is always a real number:

(a + bi) + (a - bi) = 2a

These properties are useful for simplifying expressions involving complex numbers.

Let's examine the points on the graph:

A = -2 + 3i

B = 2 + 3i

C = 3 + 2i

D =  -2 - 3i

A x D = ( -2 + 3i) x (-2 - 3i) = (-2)²- (3i)² = 4- (-9) = 13

A + D = ( -2 + 3i) +  (-2 - 3i) = -2-2 + 3i-3i = -4

Thus, points A and D are complex conjugates because the sum and product of A and D are real numbers

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The number of ways 4 people each choose a piece of cake from 7 total is 840.

Permutations are different ways of arranging objects in a definite order. It can also be expressed as the rearrangement of items in a linear order of an already ordered set. The symbol nPr is used to denote the number of permutations of n distinct objects, taken r at a time.

Given that, 4 people each choose a piece of cake from 7.

We know that, P(n,r) = n!/(n-r)!

Here, P(7, 4) =7!/(7-4)!

= (7×6×5×4×3!)/3!

= 7×6×5×4

= 840

Therefore, the number of ways 4 people each choose a piece of cake from 7 total is 840.

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The minimum amount Katie must have withheld from each of her 26 bi-weekly paychecks in order to avoid owing the federal government at the end of the year is $243.

What is income tax?

An income tax is a charge levied against people or organizations in relation to the income or profits they make. In most cases, income tax is calculated as the sum of the tax rate and the amount of taxable income. The type of taxpayer and the type of income are two factors that can affect the tax rate.

Given:

Katie has determined that, based on her annual income of $42,000, her federal income tax for the year is $6,328.

Katie gets paid on a bi-weekly basis, giving her 26 paychecks throughout the 52 week year.

We can calculate the amount that should be withheld every two weeks by dividing Katie's estimated annual income taxes by 26 weeks:

$6328 / 26 weeks = $243.38 ≈ $243.

Hence, the minimum amount Katie must have withheld from each of her 26 bi-weekly paychecks in order to avoid owing the federal government at the end of the year is $243.

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Answer:

The difference is 60.

Step-by-step explanation:

Since no more than 5 people can stay in each of the 15 rooms, we can multiply these numbers to get the largest amount of guests.

5 x 15 = 75

We assume the smallest amount can be 1 person in each of the 15 rooms.

1 x 15 = 15

We then subtract the smallest from the largest to find the difference.

75 - 15 = 60

False, Stay on SRATC3 but decrease to the point touching LRATC.

What is  LRAC curve ?

The LRAC curve is the curve that a company would follow in order to reduce its cost per unit for each specific long-run quantity of output.

              One particular level of fixed costs is represented by each of the short-run average cost (SRAC) curves that make up the LRAC curve.

Why is Lrac crucial?

Over a variety of outputs, the effect is a decrease in average costs. A company may gain a competitive edge in a market thanks to these lower expenses, which signify an increase in production efficiency.

                        Consumer prices may be reduced as a result, and corporate earnings and dividends may increase.

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Answer: D. 3/8

Step-by-step explanation:

The rate of change or slope is rise/run

We see that the y decreases by 9 and the x decreases by 24, so our slope is -9/-24 = 3/8 because negative and negative will be positive.

So the answer is D

Answer:

C. -3/8

Step-by-step explanation:

To find the rate of change is the same as finding the slope. To find the slope between two points you'd use the slope formula, which is y2-y1 over x2-x1.

You'd get -8-1 over -8-16. -8-1 is -9. -8-16 is -24 so you'd get -9/-24.

Because that isn't one of the answer choices you have to simplify it. To simplify it you would divide both numbers by the same number. A common number they both share is 3. So then you'd divide -9 by 3 and you'd divide -24 by 3. -9 divided by 3 is -3. -24 divided by 3 is -8 so you'd get -3/-8. Which should be C. -3/8. Because both numbers are negative the answer should have a negative sign so you'd cross b and d out. And a isn't right because the 8 is supposed to be in the denominator and not in the numerator.

(Hopefully you understood this, if not then I'm sorry.)

Answer:

C. 3

Step-by-step explanation:

First, use the distributive property. Multiply 4x and 1 each by 2.

This gives us the equation 50 + 8x - 2 = 72.

Next, subtract 2 from 50 to get 48. Subtract 2 from 2 to cancel it out.

Then, we have 48 + 8x = 72.

Subtract 48 from both sides. This leaves 8x = 24.

Divide both sides by 8 to get x = 3.