Unit 4 parallel and perpendicular lines unit test.

Answer:

16 [tex]\pi[/tex] cm

Step-by-step explanation:

C = 8 * 2 * pi = 16 [tex]\pi[/tex] cm

Answer:C≈50.27cm

Step-by-step explanation:

C=2πr=2·π·8≈50.26548cm

The rate of change for the given graph is 10.

Rate of change

Rate of change describes how one quantity changes in relation to another quantity.

The formula for the rate of change is,

R = Δx/Δy

where

Δx = x₂ - x₁

Δy = y₂ - y₁

where (x₁, y₁) and (x₂, y₂) refers the points given.

Given,

Here we have the graph that shows the distance, d, meters, travelled by Jaden on her bicycle as a function of the number of wheel revolutions, n, as she rode from Whitehorse to the Grey Mountain Road lookout in the Yukon.

And we need to determine the rate of change of the graph.

In order to find the rate of change , we have to select the points from the given graph.

So, here we have select the following point. (2400, 5000) and (2500, 6000)

Now, as per the rate of change formula, it can be calculated as,

Δx = 2500 - 2400

Δx = 100

And the next one is,

Δy = 6000 - 5000

Δy = 1000

Then the rate of change is,

R = 1000/100

R = 10

Therefore, the rate of change is 10.

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Answer:9

Step-by-step explanation:

Unit conversion is the conversion of one unit to another unit with its standard conversion.

The number of dimes is 5.

The number of quarters is 11.

Unit conversion is the conversion of one unit to another unit with its standard conversion.

Example:

1 minute = 60 second

1 hour = 60 minutes

1 km = 1000 m

We have,

0.1 dollar = 1 dimes

0.25 dollar = 1 quarter

Now,

D + Q = 16 _____(1)

0.1D + 0.25Q = 3.25 ______(2)

From (1) we get,

D = 16 - Q ____(3)

Putting in (2) we get,

0.1D + 0.25Q = 3.25

0.1 (16 - Q) + 0.25Q = 3.25

1.6 - 0.1Q + 0.25Q = 3.25

1.6 + 0.15Q = 3.25

0.15Q = 1.65

Q = 11

Now,

D = 16 - Q

D = 16 - 11 = 5

D = 5

Thus,

The number of dimes is 5.

The number of quarters is 11.

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Step-by-step explanation:

Given, [tex]{ \green{ \sf{y = A {e}^{mx} + B {e}^{nx}}}} [/tex]

differentiate with respect to x

this is in the form of [tex]{ \tt{ \frac{dy}{dx} ( { {e}^{a} }^{x}) = {ae}^{ax}}} [/tex]

[tex]{ \red{ \sf{ \frac{dy}{dx} = A {me}^{mx} + B {ne}^{nx}}}} [/tex]

[tex]{ \red{ \sf{ \frac{{d}^{2} y}{d{x}^{2} } = Am.{me}^{mx} + Bn. {ne}^{nx}}}} [/tex]

[tex]{ \red{ \sf{ \frac{{d}^{2} y}{d{x}^{2} } = A {m}^{2} {e}^{mx} + B {n}^{2} {e}^{nx}}}} [/tex]

L.H.S.[tex]{ = \purple{ \sf{ \frac{ {d}^{2} y}{d {x}^{2} } - (m + n) \frac{dy}{dx} + mny}}}[/tex]

[tex]{ \purple{ \sf{A {m}^{2} {e}^{mx} + B {n}^{2} {e}^{nx} - (m + n)( A {me}^{mx} + B {ne}^{nx}) \\ \\ + mn(A {e}^{mx} + B {e}^{nx})}}}[/tex]

[tex]{ \purple{ \sf{ \cancel{A {m}^{2} {e}^{mx} }} + { \cancel{B {n}^{2} {e}^{nx} }- \cancel {A{m}^{2}{e}^{mx}}} { \cancel{-B {mne}^{nx} }} - { \cancel{A {mne}^{mx}}} - { \cancel{B{n}^2 {e}^{nx}}+{ \cancel{Amn{e}^{mx}}+{ \cancel{Bmn{e}^{nx}}}}}}}[/tex]

= 0 → R.H.S

Step-by-step explanation:

I'm assuming "A" and "B" are some constants that are coefficients as well as "m" and "n". Under this assumption we can take the derivative and then take the derivative once more to find the 2nd derivative which is denoted as: [tex]\frac{d^2y}{dx^2}[/tex]

The chain rule can be denoted as: [tex]\frac{dy}{dx}=\frac{dy}{du}*\frac{du}{dx}[/tex]

It can also be denoted in function notation as: [tex]h(x)=f(g(x)) \implies h'(x)=f'(g(x))*g'(x)[/tex]

and in general it deals with composite functions and it it's going to help us take the derivative of the exponentials.

So take for example: [tex]Ae^{mx}[/tex], we can rewrite this as a composite function where: [tex]u=mx\implies Ae^{mx}=Ae^u[/tex]

So from here we can just take the derivative of [tex]Ae^u[/tex] and then multiply it by the derivative of "u". We can also apply the same thing to [tex]Be^{nx}[/tex]

So let's first take the derivative of both sides, with respect to x

[tex]\frac{d}{dx}[y]=\frac{d}{dx}[Ae^{mx} + Be^{nx}][/tex]

From here we can take the derivative of [tex]Ae^{mx} \text{ and }Be^{nx}[/tex] and then add them after taking the derivative individually.

[tex]\frac{dy}{dx}=\frac{d}{dx}[Ae^{mx}]+\frac{d}{dx}[Be^{nx}][/tex]

Assuming that "A" and "B" are constants, we can just take the derivative of: [tex]e^{mx}\text{ and }e^{nx}[/tex] and then multiply by "A" and "B" after.

[tex]\frac{dy}{dx}=A\frac{d}{dx}[e^{mx}]+B\frac{d}{dx}[e^{nx}][/tex]

now from here we apply the chain rule. I'll start with: [tex]e^{mx}[/tex]. First let's rewrite the equation so that we have a composite function: [tex]u=mx\implies e^{mx}=e^u[/tex] now from here, we can apply the chain rule: [tex]\frac{d}{dx}[e^{mx}]=\frac{d}{dx}[e^u]*\frac{d}{dx}[u][/tex]

You may know that the derivative of e to the x is just going to be e to the x, so the derivative of e^u is just e^u. Now substitute mx back in as u, and you get e^(mx). Now take the derivative of "u" which is just mx, and you get "m" since the derivative of any linear equation is just the slope (since this is constant): [tex]\frac{d}{dx}[e^{mx}]=e^{mx}*m[/tex]

Now from here we can apply the same logic to e^(nx) to get: [tex]\frac{d}{dx}[e^{nx}]=e^{nx}*n[/tex]

So now we have the following expression:

[tex]\frac{dy}{dx}=A[e^{mx}*m]+B[e^{nx}*n][/tex]

[tex]\frac{d}{dx}[\frac{dy}{dx}]=\frac{d}{dx}[A(e^{mx}*m)+B(e^{nx}*n)][/tex]

we can apply some of the rules we used before like taking each individual derivative then adding them after, and "factoring out" the A and B (because they're constants). This gives us the following expression:

[tex]\frac{d^2y}{dx^2}=A\frac{d}{dx}[e^{mx}*m]+B\frac{d}{dx}[e^{nx}*n][/tex]

From here we can also think of "factoring out" the m and n because they're constants, so we get the following:

[tex]\frac{d^2y}{dx^2}=Am*\frac{d}{dx}[e^{mx}]+Bn*\frac{d}{dx}[e^{nx}][/tex]

From here we can apply the chain rule... except we already did this work (taking the derivative of e to the mx and e to the nx) so we can just put that here

[tex]\frac{d^2y}{dx^2}=Am*[e^{mx}*m]+Bn*[e^{nx}*n][/tex]

We can further simplify this to:

[tex]\frac{d^2y}{dx^2}=Am^2*[e^{mx}]+Bn^2*[e^{nx}][/tex]

So we know that: [tex]\frac{d^2y}{dx^2}=Am^2*[e^{mx}]+Bn^2*[e^{nx}][/tex]

and that: [tex]\frac{dy}{dx}=A[e^{mx}*m]+B[e^{nx}*n][/tex] or [tex]\frac{dy}{dx}=Am[e^{mx}]+Bn[e^{nx}][/tex]

Plugging these into the expression: [tex]\frac{d^2y}{dx^2} - (m+n)(\frac{dy}{dx}})+mny=0[/tex]

we get the following: [tex]Am^2e^{mx}+Bn^2e^{nx}-(m+n)(Ame^{mx}+Bne^{nx})+mny=0[/tex]

First let's distribute the (m+n) to the first derivative to get:

[tex](m+n)(Ame^{mx}+Bne^{nx}) =(Am^2e^{mx} +Bnme^{nx} + Anme^{mx} + Bn^2e^{nx})[/tex]

so plugging this in we get:

[tex]Am^2e^{mx}+Bn^2e^{nx}-(Am^2e^{mx} +Bnme^{nx} + Anme^{mx} + Bn^2e^{nx})+mny=0[/tex]

from here we can distribute the negative sign and cancel out values to get:

[tex]-Bnme^{nx} -Anme^{mx}+mny=0[/tex]

So we're almost done and it may look a bit weird in this situation since none of the variables look like they align to cancel them out... but notice how there is a "y" variable? We know what that is equal to! We started with that expression: [tex]y=Ae^{mx}+Be^{nx}[/tex]

So let's plug this in to our expression:

[tex]-Bnme^{nx} -Anme^{mx}+mn(Ae^{nx}+Be^{nx})=0[/tex]

Let's distribute the "mn" using the distributive property:

[tex]-Bnme^{nx} -Anme^{mx}+(Anme^{nx}+Bnme^{nx})=0[/tex]

and from here we can finally cancel out all the values! and this simplifies to zero.

[tex]0=0[/tex]

so we have proven that this expression is always equal to zero!

Answer:

Step-by-step explanation:

No, buying small bag and medium bag of rubber bands will not give him more rubber bands than a large bag.

Combining things and counting them as one big group is done through addition. In math, addition is the process of adding two or more integers together. Addends are the numbers that are added, while the total refers to the outcome of the operation. The combination of at least two numbers constitutes basic addition. Once a student has mastered counting, they can begin to study basic addition. By locating the addend at the top of the chart and tracing down to the row of the second addend, they can utilize an addition chart as a learning tool.

Given,

Small = 25

Medium = 45

Large = 70

Juan wants to buy small bag and medium bag of rubber bands.

So, adding:

Small + Medium

25 + 45

70

No, buying small bag and medium bag of rubber bands will not give him more rubber bands than a large bag.

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Answer:

See below

Step-by-step explanation:

Read across the x-axis to '5' then go up to intersect the graph.....then read to the LEFT to find the price on the y-axis

Answer:

8 %

Step-by-step explanation:

To find the tax rate, we first have to find the tax

4.32 - 4 = .32

The tax rate is found by the tax / total

.32 /4 =.08

To change this to a percent multiply by 100

.08 * 100% = 8 %

If lying with the wind the same plane travels 5820 km in six hours. The  rate of the plane and still air  is 770 mph and the rate of wind is 200 mph.

Let J represent the rate of jet in still air

Let W represent the  rate of wind

Distance:

d = rt

r = d/t

Against the wind the Jet flies at a rate of  J - W

= 2850 mi/ 5hrs = 570 mph

With the wind the Jet flies at a rate of:

J+W = 5820 mi/6hrs = 970 mph

The average rate for Jet

J = (570+970)/2

J = 770 mph

So,

W = 770 - 570 = 200 mph

The rate of the plane and still air is 770 mph

The rate of wind is  200 mph.

Therefore 770 mph and 200 mph are the rate.

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Answer:

2

Step-by-step explanation:

(-11, -16) and (-13, -20)

To find the slope of the line, we use the slope formula: (y₂ - y₁) / (x₂ - x₁)

Plug in these values:

(-20 - (-16)) / (-13 - (-11))

Simplify the parentheses.

= (-20 + 16) / (-13 + 11)

= (-4) / (-2)

Simplify the fraction.

-4/-2

4/2

= 2

This is your slope.

Equation of the line - Slope

The points of the slope are, we also apply the formula:

  [tex]\huge \boxed{\boldsymbol{\sf{\left \{ {{x_1,y_1} \atop {x_2,y_2}} \right. \iff \ m=\frac{y_2-y_1}{x_2-x_1} }}}[/tex]

The points are:

     [tex]\boldsymbol{\sf{\underbrace{x_1=-11} \ \ \ \ \ \ \ \ \ \ \overbrace{y_1=-16} \ \ \ \ \ \ \ \ \ \ \underbrace{x_2=-13} \ \ \ \ \ \ \ \ \ \ \overbrace{y_{2}=-20} }}[/tex]

Substitute data into the formula:

[tex]\boldsymbol{\sf{m=\dfrac{\Delta y}{\Delta x} \iff \ m=\dfrac{y_2-y_1}{x_2-x_1} }}[/tex]

                          [tex]\boldsymbol{\sf{m=\dfrac{-20-(-16)}{-13-(-11)} }} \\ \\ \\ \boldsymbol{\sf{m=\dfrac{-4}{-2} }}\\ \\ \\ \boxed{\boxed{\boldsymbol{\sf{m=2}}}}[/tex]

The slope through the points (-11,-16) and (-13, -20) is 2.

The green family will double their investment in 34 years after the principal is subject to compound interest continuously.

The principal invested = $2500

Rate of interest annually = 2.75%

Time = t years

Amount  = double of principal = 2500 × 2 = $ 5000

Interest is compounded continuously

Formula to calculate the amount is given by:

[tex]A=Pe^{rt}[/tex]

Now we will use the values to find the exact value of t

or, [tex]5000 = 2500 e^{0.0275t}[/tex]

or, ln 2 = 0.0275t

or, t = 25.205 years

or, t ≈ 25 years

Compound interest allows interest to accumulate over time as opposed to simple interest, which does not compound because prior interest is simply not added to the principal for the current period.

The simple annual interest rate is obtained by multiplying the interest per period by the total number of periods in a year.

Hence the Green family will have doubled the investment in approximately 25 years.

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Answer:

Step-by-step explanation:

18 times 9 then -9

Answer:

135

Step-by-step explanation:

First, Times 18 x 8. which gives you 144. Then, subtract 9, giving you 135.

The inverse funciton is:

g(x) =  ∛(x/0.25)

And the graph of the two functions can be seen in the image at the end.

We want to find the inverse of the function:

f(x) = 0.25*x^3

Two functions f(x) and g(x) are inverses if and only if:

f( g(x)) = x

g(f(x)) = x

So, let's say that g(x) is the inverse of f(x), then we can write:

f(g(x)) = 0.25*(g(x))^3

And that must be equal to x, then we can solve:

0.25*(g(x))^3 = x

(g(x))^3 = x/0.25

g(x) = ∛(x/0.25)

That is the inverse fuction, the graph of the two functions can be sen below.

Where the red one is g(x) and the black one is f(x).

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The ratio of green balloons to the total number of balloons is 3 to 8.

What is Ratio?

By comparing two amounts of the same units, the ratio can be used to calculate how much of one item is included in the other. There are two different categories for ratios. The first is a part-to-part ratio, whereas the second is a part-to-whole ratio.

Given, the number of red balloons = 22

the number of blue balloons = 18

the number of green balloons = 24

Total number of balloons = 22 + 18 + 24 = 64

Now,

the ratio of green balloons to the total number of balloons =

= (the number of green balloons)/(Total number of balloons)

= 24/64

= 3/8

= 3:8

Hence, the required ratio is 3 to 8.

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Answer:

y=3/5x+3

Step-by-step explanation:

Im sorry if this is wrong your picture is a little blurry or something but

for the +3 part you go up 3 then find a pretty spot on the graph(when the line goes through a point) and  [tex]\frac{rise}{run}[/tex]  would be 3 over five since you go down 3 and to the left 5 which would both be negative but two negatives make a positive.

Hope this makes sense.:)

1 km = 1000 m and 1m = 100cm

1 km is the same as 1000 m = 1000 x 100 = 100000 cm

There are 862 kilometres so there is 862 x 100000 = 86200000 cm in 862 km.

Answer:

86200000

Step-by-step explanation:

The way to calculate confidence interval for difference in means is given below:

A range of values that, with a certain degree of confidence, are likely to represent the genuine difference between two population means is known as a confidence interval (C.I.) for a difference between means.

They will go out and take a random sample from each population, compute the mean for each sample, and use that information to estimate the difference. The difference between the two means can then be compared.

Hence, Confidence interval = (x1–x2) +/- t*√((sp2/n1) + (sp2/n2))

Example:

Let's say our goal is to determine the variation in mean weight between two turtle species, so we go out and randomly select 15 turtles from each population. The summaries for each sample are as follows:

Sample 1:

x1 = 310

s1 = 18.5

n1 = 15

Sample 1:

x2 = 300

s2 = 16.4

n2 = 15

For 90% Confidence Interval will be"

(310-300) +/- 1.70*√((305.61/15) + (305.61/15))

=  [-0.8589, 20.8589]

For 95% Confidence Interval will be:

(310-300) +/- 2.05*√((305.61/15) + (305.61/15))

=  [-3.0757, 23.0757]

For 99% Confidence Interval will be:

(310-300) +/- 2.76*√((305.61/15) + (305.61/15))

=   [-7.6389, 27.6389]

Hence, for the Difference Between Means interpretation will be: The confidence interval [-3.0757, 23.0757] has a 95% probability of containing the actual difference in mean weight between the two turtle populations.

It's possible that there is no difference in the mean weight of the turtles in these two populations because this interval contains the value "0," which indicates that it's possible.

With regard to the mean weight of the turtles in these two populations, we cannot, therefore, say with 95% confidence that there is a difference.

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The values obtained from the data in the table are as follows;

a) The percentage of boys that regularly drink Water is 32.8%

b) The proportion of students that were Girls ≈ 0.43

c) The marginal frequency for Lemonade ≈ 74

d) The joint frequency for Girls  who regularly drink Lemonade = 32

e) The joint relative frequency for Boys who regularly drink Soda ≈ 0.21

f) The marginal relative frequency for Girls ≈ 0.43

A two-way table provides the values of the frequencies for two variables, such that one category of variables are presented using the rows and the other category of variables are presented along the columns.

a) The total number of boys = 45 + 50 + 42 = 137

The number of boys that regularly drink water = 45

The percentage of boys that regularly drink water is therefore;

[tex]Percentage = \dfrac{45}{137} \times 100 \approx 32.8\%[/tex]

b) The total number of students = 77 + 88 + 74 = 239

The number of girls = 32 + 38 + 32 = 102

The proportion of students that were girls is therefore;

[tex]\dfrac{102}{239} \approx 0.43[/tex]

c) The marginal frequency for Lemonade which is the row total for Lemonade is 74

d) The joint frequency for Girls who regularly drink Lemonade from the table is 32

e) The joint relative frequency for the Boys who regularly drink Soda is found using the equation;

[tex]JRF = \dfrac{50}{239} \approx 0.21[/tex]

f) The marginal relative frequency for the girls is given by the equation;

[tex]MRF = \dfrac{102}{239} \approx 0.43[/tex]

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The greatest possible whole number perimeter of the triangle is  29 meters .

The two sides of the triangle are 8.1 m and 6.5 m.

Now , from the properties of triangles , we can say that the third side will be less than the sum of the other two sides.

if third side is x , then we can write the inequality

x < 8.1+6.5

or, x < 14.6 m

Takin x =14.6 m. The perimeter of the triangle will be

8.1 m +6.5 m+14.6 = 29.2

Hence the greatest possible side will be 29m.

Any two-dimensional figure's perimeter is dependent on the space surrounding it. Practically any closed shape's perimeter may be calculated simply summing the lengths of its sides.

The formula used to determine the perimeter of a closed shape figure is often proportionate to the length of the figure's outside line. A triangle's perimeter is determined by multiplying its three sides together.

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The equation, 4x-6y = -24, after simplification is 2x-3y = -12.

According to the question,

We have the following equation:

4x-6y = -24

Now, this equation can easily simplified by following the given steps.

First, we will divide both sides of the equation by 2. Now, we have the following expression:

2x-3y = -24/2

2x-3y = -12

Now, this equation can not be further simplified. What we can do in this equation is to only change the sides of the terms from right hand side to the left hand side or left hand side to the right hand side.

Hence, the equation, 4x-6y = -24, after simplification is 2x-3y = -12.

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Answer:

880 mango trees and 440 star apple strees

Step-by-step explanation:

when for every 8 mango trees there are 4 star apple trees, then there is a 2:1 ratio of mango : star apple trees.
1320 divided by 3 = 440
and 2*440 = 880 mango trees, 1*440 = 440 star apple trees

(B) 1/2,1/4,1/10,6/3 are considered benchmark fractions.

Benchmark fractions are regular fractions that we can use to compare or judge other fractions when measuring, comparing, or ranking them. Benchmark fractions are simple to see and recognize, making it easier to estimate the components. We can either make the denominators of two fractions with distinct numerators and denominators the same, or we can compare them to a benchmark fraction like 1/2. The best way to use benchmark fractions is to plot the fractions to be compared on a number line next to the benchmarks.

A ruler is a tool most frequently used for number lines. As reference points, it uses half, fourth, and eighth.

Only option (B) satisfies the meaning of the benchmark fraction.

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The rocket reaches its maximum height at 2.25 second(s) after launch

The maximum height reached by the object is 88 feet.

The function is given as

h(t) = -16t² + 72t + 7

Differentiate the above function

So, we have the following representation

h'(t) = -32t + 72

Set to 0

-32t + 72 = 0

So, we have

32t = 72

Divide by 32

t = 2.25

Hence, the time is 2.25 seconds

In (a), we have

t = 2.25

Substitute t = 2.25 in h(t) = -16t² + 72t + 7

h(2.25) = -16(2.25)² + 72(2.25) + 7

Evaluate

h(2.25) = 88

Hence, the maximum height is 88 feet

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Answer:

4x25/100=1

Step-by-step explanation:

How I came up with it

4÷100=25

25÷25=1

Julie will only get a 1% discount

Hope it helps

Angie' s grandma drive 376 miles in each day.

What is Division method?

Division method is used to distributing a group of things into equal parts. Division is just opposite of multiplications.

For example, dividing 20 by 2 means splitting 20 into 2 equal groups of 10.

Given that;

The two cities are 752 travel miles apart.

And, It take 12 hours to drive that far.

Now,

Since, Angie' s grandma want to do the drive in two day.

So, The distance cover by Angie' s grandma in each day is,

= 752 / 2

= 376 miles

Thus, Angie' s grandma drive 376 miles in each day.

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Answer:

Step-by-step explanation:

it’s 5.5

The zero of the function in the given table is -1, indicating that the function provided has an x-intercept.

The x-intercept is defined as an intercept that is located at the x-axis of the plane, which is the location or coordinates from where the line crosses.

The table is given in the question,

If the value of x is -1, then the value of y is zero, according to the table. This is the function's x-intercept. The zero of the function depicted in this table is -1, which signifies that the supplied function has an x-intercept.

Therefore, the zero of the function given in this table is -1, indicating that the function provided has an x-intercept.

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The amount in the fund immediately after the last deposit is $1148830.185

Given,

The amount deposited by Steve annually =  $53,700

Interest rate of the fund = 15%

Time period of the investment = 10 years

We have to find the amount in the fund immediately after the last deposit.

Here,

To calculate the future value after the last deposit, we need to use the following formula:

Future Value = {A[(1+i)^n-1]}/i

A = Annual deposit

Then,

FV = {53,700[(1.15)^9 - 1]}  / 0.15 + 53,700

FV= 1095130.185 + 53,700

FV= $1148830.185

Therefore,

The amount in the fund immediately after the last deposit is $1148830.185

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