Truck brakes can fail if they get too hot. In some mountainous areas, ramps of loose gravel are constructed to stop runaway trucks that have lost their brakes. The combination of a slight upward slope and a large coefficient of rolling friction as the truck tires sink into the gravel brings the truck safely to a halt. Suppose a gravel ramp slopes upward at 6.0∘ and the coefficient of rolling friction is 0.30. How long the ramp should be to stop a truck of 15000 kg having a speed of 35 m/s.

Answer:

  a) There is no algebraic method for finding θ in terms of n

  b) should be a = r(1 -cos(θ))

Step-by-step explanation:

Algebraic methods have been developed for solving trig functions and polynomial functions individually, but not in combination. In general, the solution is easily found numerically, but not analytically.

You would be looking for the numerical solution to ...

  f(n, θ) = 0

where f(n, θ) can be ...

  f(n, θ) = θ - (1/2)sin(2θ) - π/n

___

The attached shows Newton's method iterative solutions for n = 3 through 6:

  for n = 3, θ ≈ 1.3026628373

  for n = 4, θ ≈ 1.15494073001

...

Answer: 478

Step-by-step explanation:

The formula to calculate the standard error of the population mean is given by :-

[tex]S.E.=\dfrac{\sigma}{\sqrt{n}}[/tex], where [tex]\sigma[/tex] is the standard deviation and 'n' is the sample size.

Given: Mean : [tex]\mu=$\50,000[/tex]

Standard deviation : [tex]\sigma= $\4,000[/tex]

Sample size : [tex]n=70[/tex]

Now, the value of the standard error of the average annual salary is given by :-

[tex]S.E.=\dfrac{50000}{\sqrt{70}}=478.091443734\approx478[/tex]

Hence, the standard error of the average annual salary = 478

The value of the standard error of the average annual salary is calculated using the formula SE = population standard deviation / [tex]\sqrt{sample size}[/tex] ,which for a population standard deviation of $4,000 and a sample size of 70 employees, comes out to approximately $478.

To calculate the standard error of the average annual salary, you'd use the formula for the standard error of the mean when you know the population standard deviation: SE = σ /[tex]\sqrt{n}[/tex], where σ is the population standard deviation and n is the sample size.

In this case, the population standard deviation σ is $4,000, and the sample size n is 70 employees. Using the formula, we get SE = 4000 / [tex]\sqrt{70}[/tex], which will give us the standard error of the average annual salary.

Performing the calculation: SE = 4000 / [tex]\sqrt{70}[/tex] ≈ 4000 / 8.367 = 478.29, which when rounded to the nearest integer is $478. Therefore, the value of the standard error of the average annual salary is approximately $478.

Answer: 0.8021

Step-by-step explanation:

The given problem is a binomial distribution problem, where

[tex]n=14,\ p=0.2, q=1-0.2=0.8[/tex]

The formula of binomial distribution is :-

[tex]P(X=r)=^{n}C_{r}p^{r}q^{n-r}[/tex]

The probability that the lot will fail inspection is given by :_

[tex]P(X\geq2)=1-(P(X\leq1))\\\\=1-(P(0)+P(1))\\\\[/tex]

[tex]=1-(^{14}C_{0}(0.2)^{0}(0.8)^{14-0}+^{14}C_{1}(0.2)^{1}(0.8)^{14-1})\\\\=1-((1)(0.8)^{14}+(14)(0.2)(0.8)^{13})\\\\=0.802087907\approx0.8021[/tex]

Hence, the required probability = 0.4365

Answer:

  The bottleneck of this process is Workstation C, at 30 minutes per unit.

Step-by-step explanation:

The throughput of each workstation is ...

Since each process must be executed once per finished product, the bottleneck is the station with the lowest throughput. It is clearly Workstation C.

The bottleneck of this process is workstation D, where the time per unit is 15 minutes.

The bottleneck of this process is workstation D, where the time per unit is 15 minutes. This means that workstation D takes the longest time to complete one unit compared to other workstations. The bottleneck determines the maximum output rate of the entire process, as the other workstations cannot work faster than the slowest one.

https://brainly.com/question/32721320

#SPJ3

Answer:

A. H(x) is an inverse of F(x)

Step-by-step explanation:

The given functions are:

[tex]F(x)=\sqrt{x-2}[/tex]

[tex]G(x)=(x-2)^2[/tex]

[tex]H(x)=x^2+2[/tex]

We compose F(x) and G(x) to get:

[tex](F\circ G)(x)=F(G(x))[/tex]

[tex](F\circ G)(x)=F((x-2)^2)[/tex]

[tex](F\circ G)(x)=\sqrt{(x-2)^2-2}[/tex]

[tex](F\circ G)(x)=\sqrt{x^2-4x+4-2}[/tex]

[tex](F\circ G)(x)=\sqrt{x^2-4x+2}[/tex]

[tex](F\circ G)(x)\ne x[/tex]

Hence G(x) is not an inverse of F(x).

We now compose H(x) and G(x).

[tex](F\circ H)(x)=F(H(x))[/tex]

[tex](F\circ H)(x)=F(x^2+2)[/tex]

[tex](F\circ H)(x)=\sqrt{x^2+2-2}[/tex]

We simplify to get:

[tex](F\circ H)(x)=\sqrt{x^2}[/tex]

[tex](F\circ H)(x)=x[/tex]

Since [tex](F\circ H)(x)=x[/tex], H(x) is an inverse of F(x)

Note that [tex]x^{\frac{1}{2}}=\sqrt[2]{x}[/tex]

Which means that:

[tex]512^{\frac{1}{2}}=\sqrt[2]{512}=\sqrt[2]{16^2\cdot2}=\boxed{16\sqrt[2]{2}}[/tex]

the answer is B.

Hope this helps.

r3t40

ANSWER

[tex]80 {ms}^{ - 1} [/tex]

EXPLANATION

The average velocity of the ball is the rateof displacement over the total time.

The height of the ball, in feet, after t seconds is given by the function:

[tex]f(t)=-16t^2+192t[/tex]

At time t=1, the height of the ball is

[tex]f(1)=-16(1)^2+192(1)[/tex]

[tex]f(1)=-16+192 = 176ft[/tex]

At time t=6, the height of the ball is

[tex]f(6)=-16(6)^2+192(6)[/tex]

[tex]f(6)=-16(36)+192(6)[/tex]

[tex]f(6)=-576+1152 = 576[/tex]

The average velocity

[tex] = \frac{f(6) - f(1)}{6 - 1} [/tex]

[tex]= \frac{576- 176}{6 - 1} [/tex]

[tex]= \frac{400}{5} [/tex]

[tex] = 80 {ms}^{ - 1} [/tex]

Answer:

The area of this composite figure is [tex]64\ cm^{2}[/tex]

Step-by-step explanation:

we know that

If the composite figure is divided into two congruent trapezoids, then the area of the composite figure is equal to the area of one trapezoid multiplied by two

so

The area of the composite figure is

[tex]A=2[\frac{1}{2}(b1+b2)h][/tex]

[tex]A=(b1+b2)h[/tex]

substitute the values

[tex]A=(6+10)4[/tex]

[tex]A=64\ cm^{2}[/tex]

Answer:

64cm

Step-by-step explanation:

I looked at the guy's answer on top of mine and it was correct, go thank him!

Answer:

cool

Step-by-step explanation:

Answer:

the probability that the woman selected does not have​ red/green color​ blindness is 0.9936.

Step-by-step explanation:

Final answer:

The probability that a randomly selected woman does not have red/green color blindness is 99.36%.

Explanation:

If the rate of red/green color blindness among a certain group of women is 0.64%, this means that out of every 100 women, 0.64 women on average would have red/green color blindness.

The complement of a probability event occurring is equal to 1 minus the probability of the event.

Therefore, the probability that a randomly selected woman does not have red/green color blindness is :

1 - 0.0064

which is 0.9936 or 99.36%.

Answer:

$5151.42

Step-by-step explanation:

The formula you need is

[tex]A(t)=P(1+\frac{r}{n})^{(n)(t)}[/tex]

where A(t) is the amount after the compounding, P is the initial investment, r is the interest rate in decimal form, n is the number of compoundings per year, and t is time in years.  The info we have is

A(t) = 8000

P = ?

r = .09

t = 5

Filling in we have

[tex]8000=P(1+\frac{.09}{2})^{(2)(5)}[/tex]

Simplifying a bit and we have[tex]8000=P(1+.045)^{10}[/tex]

Now we will add inside the parenthesis and raise 1.045 to the 10th power to get

8000 = P(1.552969422)

Divide away the 155... on both sides to solve for P.

P = $5151.42

Answer: 0.01708

Step-by-step explanation:

Given : If records indicate that 15 houses out of 1000 are expected to be damaged by fire in any year.

i.e. the probability that house damaged buy fire in a year : [tex]p=\dfrac{15}{1000}=0.015[/tex]

The formula for binomial distribution is given by :-

[tex]^{n}C_xp^x(1-p)^{n-x}[/tex]

Now, the probability that a woman who owns 14 houses will have fire damage in 2 of them in a year (put n=14 and x=2), we get

[tex]^{14}C_2(0.015)^2(1-0.015)^{14-2}\\\\=\dfrac{14!}{2!(14-2)!}(0.015)^2(0.985)^{12}\\\\=0.0170788520518\approx0.01708[/tex]

Hence, the required probability = 0.01708

Answer:

Part 1) [tex]y=80,000(1.035)^{x}[/tex]

Part 2) The table in the attached figure

Part 3) The graph in the attached figure

Step-by-step explanation:

Part 1) Find the population function

In this problem we have a exponential function of the form

[tex]y=a(b)^{x}[/tex]

where

y ----> is the population

x ----> the time in years

a is the initial value (a=80,000 people)

b is the base (b=100%+3.5%=103.5%=1.035)

substitute

[tex]y=80,000(1.035)^{x}[/tex]  

Part 2) Construct the table

For x=0 years  

substitute in the function equation

[tex]y=80,000(1.035)^{0}=80,000\ people[/tex]  

For x=10 years

substitute in the function equation

[tex]y=80,000(1.035)^{10}=112.848\ people[/tex]

For x=20 years

substitute in the function equation

[tex]y=80,000(1.035)^{20}=159,183\ people[/tex]

For x=40 years

substitute in the function equation

[tex]y=80,000(1.035)^{40}=316,741\ people[/tex]

For x=50 years

substitute in the function equation

[tex]y=80,000(1.035)^{50}=446,794\ people[/tex]

For x=75 years

substitute in the function equation

[tex]y=80,000(1.035)^{75}=1,055,884\ people[/tex]

For x=100 years  

substitute in the function equation

[tex]y=80,000(1.035)^{100}=2,495,313\ people[/tex]

Part 3) The graph in the attached figure  

Answer:

x=0

Step-by-step explanation:

You are given the function [tex]f(x)=e^x.[/tex] To find it inverse function, express x in terms of y:

[tex]y=e^x\\ \\\ln y=x[/tex]

Now change x into y and y into x:

[tex]y=\ln x\\ \\f^{-1}(x)=\ln x[/tex]

The graph of the function [tex]f^{-1}(x)[/tex] has vertical asymptote x=0.

Vertical asymptote at x = 0

Answer:

Step-by-step explanation:

The forward speed of the center of the tire with respect to the ground is the same as the tangential speed of the tire at its full radius of 0.315 m, relative to the axle.

The angular speed of the tire is the ratio of tangential speed to radius:

  (19.3 m/s)/(0.315 m) ≈ 61.27 radians/s

The tangential speed at any other radius is the product of angular speed and radius. At a radius of 0.193 m, the tangential speed is ...

  (0.193 m)×(19.3 m/s)/(0.315 m) ≈ 11.825 m/s ≈ 11.8 m/s

(a) The angular speed of the wheel is approximately 61.27 rad/s.

(b) The tangential speed of a point 0.193 m from the axle is about 11.82 m/s, relative to the axle.

let's break this down step by step.

Given:

Radius of the tire, r = 0.315 m

Linear speed of the center of the tire, v = 19.3 m/s

Distance from the axle to the point, d = 0.193 m

(a) To determine the angular speed of the wheel (ω), we can use the formula relating linear speed (v) and angular speed (ω) for a rotating object:

v = ω * r

where:

v = linear speed

ω = angular speed

r = radius

We can rearrange this equation to solve for ω:

ω = v / r

Now, substitute the given values:

ω = 19.3 m/s / 0.315 m

ω ≈ 61.27 rad/s

So, the angular speed of the wheel is approximately 61.27 rad/s.

(b) To find the tangential speed of a point located 0.193 m from the axle relative to the axle, we'll use the formula:

Tangential speed (vt) = ω x distance from the axle (d)

We already have the value of ω from part (a), which is approximately 61.27 rad/s. Now, let's calculate the tangential speed:

vt = 61.27 rad/s x  0.193 m

vt ≈ 11.82 m/s

So, the tangential speed of a point located 0.193 m from the axle, relative to the axle, is approximately 11.82 m/s.

Answer:

Function which represents the product of these two numbers is:

(23+x)(28+2x)

Step-by-step explanation:

The first number is the sum of 23 and x

i.e. First number=23+x

The second number is 18 less than two times the first number.

i.e. Second number=2(23+x)-18

                                = 46+2x-18

                                = 28+2x

Product of the two numbers=(23+x)(28+2x)

Hence, function which represents the product of these two numbers is:

(23+x)(28+2x)

Answer:

3rd option: 60 degrees

Step-by-step explanation:

We can see in the diagram that the angle on C is a supplementary angle, which means that the sum of 135 and internal angle will be equal to 180 degrees.

Let x be the internal angle,

Then

x+135 = 180

x = 180-135

x = 45 degrees

So now we know that two interior angles of the triangle.

Also we know that sum of all internal angles of triangle is 180 degrees.

Using the same postulate:

A+B+C = 180

75 + B + 45 = 180

120+B = 180

B = 180 - 120

B = 60 degrees

So,

third option is the correct answer ..

Answer:

It’s 120 I got it right on the test !

Answer:  (d) 0.3558.

Step-by-step explanation:

We know that the standard error of sample mean difference is given by:-

[tex]S.E.=\sqrt{\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2}}[/tex]

Given : [tex]n_1= 20\ ,\ n_2=20[/tex]

[tex]s_1=1.10\ ,\ \ s_2=1.15[/tex]

Then , the standard error of the sampling distribution of the sample mean difference [tex]\overline{x_1}-\overline{x_2}[/tex] is equal to :-

[tex]S.E.=\sqrt{\dfrac{1.10^2}{20}+\dfrac{1.15^2}{20}}\\\\\Rightarrow\ S.E.=0.355844066973\approx0.3558[/tex]

Hence, the standard error of the sampling distribution of the sample mean difference [tex]\overline{x_1}-\overline{x_2}[/tex] is equal to 0.3558.

Final answer:

The standard error of the sampling distribution of the sample mean difference is calculated using the formula involving standard deviations and sample sizes of the independent samples; the correct answer, after computation, is 0.3558.

Explanation:

The standard error of the sampling distribution of the sample mean difference (ëxbar1 - ëxbar2) when assuming population variances are equal can be computed using the formula for the standard error of the difference of two independent sample means, which is the square root of the sum of their variances divided by their respective sample sizes. The formula is:
SE = √((s1²/n1) + (s2²/n2))

Given the summary statistics:

The calculation of the standard error would be:

SE = √((1.10²/20) + (1.15²/20))

SE = √((1.21/20) + (1.3225/20))

SE = √(0.0605 + 0.066125)

SE = √(0.126625)

SE = 0.3558 (when rounded to four decimal places)

Hence, the correct answer is option (d) 0.3558.

Answer:

The area under the curve y=f(x) on [a,b] is [tex]\frac{1}{6}\ln(2)[/tex] square units.

Step-by-step explanation:

The given function is

[tex]f(x)=\tan(3x)[/tex]

where a=0 and b=pi/12.

The area under the curve y=f(x) on [a,b] is defined as

[tex]Area=\int_{a}^{b}f(x)dx[/tex]

[tex]Area=\int_{0}^{\frac{\pi}{12}}\tan (3x)dx[/tex]

[tex]Area=\int_{0}^{\frac{\pi}{12}}\frac{\sin (3x)}{\cos (3x)}dx[/tex]

Substitute cos (3x)=t, so

[tex]-3\sin (3x)dx=dt[/tex]

[tex]\sin (3x)dx=-\frac{1}{3}dt[/tex]

[tex]a=\cos (3(0))=1[/tex]

[tex]b=\cos (3(\frac{\pi}{12}))=\frac{1}{\sqrt{2}}[/tex]

[tex]Area=-\frac{1}{3}\int_{1}^{\frac{1}{\sqrt{2}}}\frac{1}{t}dt[/tex]

[tex]Area=-\frac{1}{3}[\ln t]_{1}^{\frac{1}{\sqrt{2}}[/tex]

[tex]Area=-\frac{1}{3}(\ln \frac{1}{\sqrt{2}}-\ln (1))[/tex]

[tex]Area=-\frac{1}{3}(\ln 1-\ln \sqrt{2}-0)[/tex]

[tex]Area=-\frac{1}{3}(-\ln 2^{\frac{1}{2}})[/tex]

[tex]Area=-\frac{1}{3}(-\frac{1}{2}\ln 2)[/tex]

[tex]Area=-\frac{1}{6}\ln 2[/tex]

Therefore the area under the curve y=f(x) on [a,b] is [tex]\frac{1}{6}\ln(2)[/tex] square units.

Answer:

Step-by-step explanation:

Given:

  Total interest earned on two investments is $29,220.

  An amount is invested at 7%.

  $24,000 more than 4 times that amount is invested at 8%.

Find:

  The amount invested at each rate.

Solution:

  Our strategy will be to define a variable representing the amount invested at 7%, use that variable to write an expression for the amount invested at 8%, then write an equation for the total return on the investments.

Let x represent the amount invested at 7%. Then (24,000+4x) will be the amount invested at 8%. The total interest earned will be ...

  interest on 7% account + interest on 8% account = total interest

  0.07x + 0.08(24000+4x) = 29220

  0.39x + 1920 = 29220 . . . . . . . . . . . simplify

  0.39x = 27300 . . . . . . . . . . . . . . . . . .subtract 1920

  x = 27300/0.39 = 70000 . . . . . . . . . divide by the coefficient of x

  24,000 +4x = 24,000 +280,000 = 304,000 . . . . amount invested at 8%

He invested $70,000 at 7% and $304,000 at 8%.

Check

The answer must satisfy ...

  7% interest + 8% interest = 29,220

  0.07×70,000 +0.08×304,000 = 4,900 +24,320 = 29,220 . . . . as required

_____

Comment on 6-step method

We have tried to hit the highlights. Your steps appear to be ...

Answer:

F or G = {7,8,9,10,11,12,13,14}

n(F or G) = 8

n(S) = 12

By counting the no. of outcome

P(F or G) = n(F or G) / n(S)

P(F or G) = 8 /12

P(F or G) = 2/3

By using the general addition rule

P(F or G) = P(F) + P(G) - P(F and G)

= 6/12 + 4/12 - 2/12

= 2/3

To find the number of people who did not like any of the three flavors, we need to subtract the number of people who liked at least one flavor from the total number of people.

To find the number of people who did not like any of the three flavors, we need to subtract the number of people who liked at least one flavor from the total number of people.

From the given information, we can create a Venn diagram to represent the preferences:

Picking it up from the explanation above, it becomes clear that 10 people liked both vanilla and strawberry, 10 people liked both strawberry and chocolate, and 15 people liked both vanilla and chocolate. We also know that 5 people liked all three flavors. Using this information, we can determine the number of people who liked at least one flavor by adding up the numbers in the overlapping circles: 10 + 10 + 15 + 5 = 40 people.

To find the number of people who did not like any of the three flavors, we subtract 40 from the total number of people who responded to the survey: 100 - 40 = 60 people.

https://brainly.com/question/18427332

#SPJ3

To find out how many people did not like any of the three flavors, we use the principle of inclusion-exclusion, resulting in 5 people who did not like vanilla, chocolate, or strawberry.

The student has asked a question related to combinations without repetition and the interpretation of survey results. To determine how many people did not like any of the three flavors (vanilla, chocolate, strawberry), we can use the principle of inclusion-exclusion. Here are the steps to solve this:

Therefore, 5 people did not like any of the three flavors.

Answer:

a) 43

b) 5.7

c) 2.54

Step-by-step explanation:

The metric system is so easy to work with, everything is in base of 10, and all measures use a prefix. Here, we're talking about length, with the main measure being the meter.

1 m = 100 centimeters (centi- = 1/100)

1 m = 1000 millimeters (milli- = 1/1000)

So, for each centimeter, you have 10 millimeters.

a) How many millimeters are in 4.3 centimeters?

1 cm = 10 mm as shown above, so 4.3 cm = 43 mm

b) How many centimeters are in 57 millimeters?

10 mm = 1 cm so, 57 mm = 5.7 cm

c) Approximately how many centimeters are in an inch?

There are approximately 2.54 cm in an inch.  No real calculation to make here, it's just a unit conversion between systems, done following a known reference table.

Final answer:

Lester Hollar can assess the impact of the fitness program on absenteeism by conducting a paired samples t-test at a 0.05 significance level. If the p-value is less than 0.05, the program effectively reduced absences; otherwise, there is insufficient evidence to conclude its effectiveness.

Explanation:

Evaluating the Exercise Program’s Impact on Employee Absenteeism

Lester Hollar wishes to assess if the company’s fitness program led to a decline in employee absences. With a sample of eight participants, he analyzed absenteeism before and after the program’s implementation. To determine if absences have decreased, a hypothesis test at the 0.05 significance level (alpha) is conducted.

To evaluate the change in absences, two sets of absenteeism data are compared using a statistical test, such as the paired samples t-test. This test examines if the mean difference in absences before and after the program is statistically significant. If the p-value obtained from the test is less than the significance level of 0.05, the null hypothesis (no change in absences) would be rejected, suggesting that the exercise program was effective in reducing absences.

If Lester Hollar finds a p-value greater than 0.05, he would not reject the null hypothesis, indicating that there isn’t sufficient evidence to conclude the program's impact. It’s also important to estimate the p-value precisely as it gives a measure of the strength of evidence against the null hypothesis. However, without the specific data, we cannot calculate the p-value or make a definitive conclusion here.

Answer:

G.P. is, 450, 360, 288, 230.4,......

The sum is 2250.

Step-by-step explanation:

Given,

The first total spending in the resort city = $ 450 million,

Also, 80% of that revenue is again spent in the resort city, and of that amount approximately 80% is again spent in the same city, and so on.

Thus, there is a G.P. that represents the given situation,

450, 360, 288, 230.4,......

Which is an infinite geometric series having first term, a = 450,

Common ratio, r = 0.8,

Hence, the sum of the series,

[tex]S_n=\frac{a}{1-r}[/tex]

[tex]=\frac{450}{1-0.8}[/tex]

[tex]=\frac{450}{0.2}[/tex]

[tex]=2250[/tex]

Answer:

Yes, the farmer have enough paint to complete the​ job.

Step-by-step explanation:

It is given that a gas storage tank is in the shape of a right circular cylinder.

The radius of the base is 2 ft and the height of cylinder is 3 ft.

The total surface area of a cylinder is

[tex]S=2\pi rh+2\pi r^2[/tex]

Total surface area of gas storage tank is

[tex]S=2\pi (2)(3)+2\pi (2)^2[/tex]

[tex]S=12\pi+8\pi[/tex]

[tex]S=20\pi[/tex]

[tex]S=62.8318530718[/tex]

[tex]S\approx 62.83[/tex]

The total surface area of gas storage tank is 62.83 square feet.

The farmer has 1 gallon of paint and 1 gallon of paint will cover 162 square​ feet.

Since 62.83<162, therefore 1 gallon of paint is enough to paint the gas storage.

Hence the required statement is Yes, the farmer have enough paint to complete the​ job.

Answer:

Yes, the farmer have enough paint to complete the​ job.

Step-by-step explanation:

1 gallon is good

height: 6

formula: 1/2bh

hope this helps :)

For this case we have that by definition, the area of a triangle is given by:

[tex]A = \frac {1} {2} b * h[/tex]

Where:

b: It's the base

h: It's the height

They tell us as data that:

[tex]A = 24 \ in ^ 2\\b = 8in[/tex]

Substituting the data and clearing the height:

[tex]24 = \frac {1} {2} 8 * h\\24 = 4h\\h = \frac {24} {4}\\h = 6[/tex]

So, the height of the triangle is 6in

Answer:

[tex]h = 6in[/tex]

Answer:

[tex]y=4.5(0.5)^{x}[/tex]

Step-by-step explanation:

* Lets revise the meaning of exponential function

- The form of the exponential function is [tex]y=ab^{x}[/tex],

  where a ≠ 0, b > 0 ,  b ≠ 1, and x is any real number

- It has a constant base b

- It has a variable exponent x

- To solve an exponential equation, take the log or ln of both sides,  

 and solve for the variable

* Lets solve the problem

∵ y = a(b)^x is an exponential function

∵ Its graph contains the point (-4 , 72) and (-2 , 18)

- Lets substitute x and y by the coordinates of these points

# Point (-4 , 72)

∵ [tex]y=ab^{x}[/tex]

∵ x = -4 and y = 72

∴ [tex]72=ab^{-4}[/tex]

- The change any power from -ve to +ve reciprocal the base of

 the power ([tex]p^{-n}=\frac{1}{p^{n}}[/tex]

∴ [tex]72=\frac{a}{b^{4}}[/tex]

- By using cross multiplication

∴ [tex]a=72b^{4}[/tex] ⇒ (1)

# Point (-2 , 18)

∵ x = -2 and y = 18

∴ [tex]18=ab^{-2}[/tex]

∴ [tex]18=\frac{a}{b^{2}}[/tex]

- By using cross multiplication

∴ a = 18b² ⇒ (2)

- Equate the two equations (1) and (2)

∴ [tex]72b^{4}=18b^{2}[/tex]

- Divide both sides by 18b²

∵ [tex]\frac{72b^{4}}{18b^{2}}=4b^{4-2}=4b^{2}[/tex]

∵ [tex]\frac{18b^{2}}{18b^{2}}=(1)b^{2-2}=(1)b^{0}=(1)(1)=1[/tex]

∴ 4b² = 1 ⇒ divide both sides by 4

∴ [tex]b^{2}=\frac{1}{4}=0.25[/tex] ⇒ take square root for both sides

∴ b = √0.25 = 0.5

- Lets substitute the value ob b in equation (1) or (2) to find a

∵ a = 18b²

∵ b² = 0.25

∴ a = 18(0.25) = 4.5

- Lets substitute the values of a and b in the equation [tex]y=ab^{x}[/tex]

∴ [tex]y=4.5(0.5)^{x}[/tex]

- We can write it using fraction

∴ [tex]y=\frac{9}{2}(\frac{1}{2})^{x}[/tex]

ANSWER

[tex]y = \frac{9}{2} ( { \frac{1}{2} })^{x}[/tex]

EXPLANATION

Let the exponential function be

[tex]y = a {b}^{x} [/tex]

Since the graph includes (–4, 72), it must satisfy this equation.

[tex]72= a { b}^{ - 4}[/tex]

Multiply both sides by b⁴ .

This implies that,

[tex]a = 72 {b}^{4} ...1[/tex]

The graph also includes (-2,18).

We substitute this point also to get:

[tex]18=a {b}^{ - 2} [/tex]

Multiply both sides by b²

[tex]a = 18 {b}^{2} ...(2)[/tex]

We equate (1) and (2) to obtain:

[tex]72 {b}^{4} = 18 {b}^{2} [/tex]

Multiply both sides by

[tex] \frac{72 {b}^{4} }{ {18b}^{4} } = \frac{18 {b}^{2} }{18 {b}^{4} } [/tex]

[tex]4 = \frac{1}{ {b}^{2} } [/tex]

Or

[tex]{2}^{ 2} = ( \frac{1}{b} )^{2} [/tex]

[tex] \frac{1}{b} = 2[/tex]

[tex]b = \frac{1}{2} [/tex]

Put b=½ into equation (2).

[tex]a = 18 {( \frac{1}{2} })^{2} [/tex]

[tex]a = \frac{18}{4} [/tex]

[tex]a = \frac{9}{2} [/tex]

Therefore the equation is

[tex]y = \frac{9}{2} ( { \frac{1}{2} })^{x} [/tex]

Answer:

First type of fruit drinks: 48 pints

Second type of fruit drinks: 32 pints

Step-by-step explanation:

Let's call A the amount of first type of fruit drinks. 55% pure fruit juice

Let's call B the amount of second type of fruit drinks. 80% pure fruit juice

The resulting mixture should have 65% pure fruit juice and 80 pints.

Then we know that the total amount of mixture will be:

[tex]A + B = 80[/tex]

Then the total amount of pure fruit juice in the mixture will be:

[tex]0.55A + 0.8B = 0.65 * 80[/tex]

[tex]0.55A + 0.8B = 52[/tex]

Then we have two equations and two unknowns so we solve the system of equations. Multiply the first equation by -0.8 and add it to the second equation:

[tex]-0.8A -0.8B = -0.8*80[/tex]

[tex]-0.8A -0.8B = -64[/tex]

[tex]-0.8A -0.8B = -64[/tex]

               +

[tex]0.55A + 0.8B = 52[/tex]

--------------------------------------

[tex]-0.25A = -12[/tex]

[tex]A = \frac{-12}{-0.25}[/tex]

[tex]A = 48\ pints[/tex]

We substitute the value of A into one of the two equations and solve for B.

[tex]48 + B = 80[/tex]

[tex]B = 32\ pints[/tex]

To create an 80-pint batch of 65% pure fruit juice, the Royal Fruit Company needs to solve two equations representing the volume and percent mixture of the two juices. These equations can be solved simultaneously to find the required volumes of each juice.

The subject of this question falls under Mathematics, particularly dealing with proportions and algebra. Given that the first type of juice is 55% pure fruit and the second type is 80% pure fruit, we can define our variables: let's denote X as the volume of the first type of drink and Y as the volume of the second one. We know that the total volume is 80 pints, so we have our first equation: X + Y = 80. The second equation derives from the percentage of fruit juice: 0.55X + 0.80Y = 0.65*80.

Now we can solve these two equations to find the volumes of X and Y. The solution to these equations will provide us with the volume needed from each of the two types of juice to achieve a 65% pure fruit juice drink.

https://brainly.com/question/24875240

#SPJ3

Answer:

The area of the glass in region BCIH is 11 to the nearest feet²

Step-by-step explanation:

* Lets explain the figure

- The window is a semicircle with center G and radius 6 feet

- There is a small semicircle with center G and radius GF

∵ GE is 6 feet and EF is 3 feet

∵ GE = GF + FE

∴ 6 = GF + 3 ⇒ subtract 3 from both sides

∴ 3 = GF

∴ The radius of the small semicircle is 3 feet

∵ m∠BGC = 45°

- The area of sector BGC is part of the area of the semicircle

∵ The area of semi-circle is 1/2 π r²

∵ The measure of the central angle of the semicircle is 180°

∵ The measure of the central angle of the sector BGC is 45°

∴ The sector = 45°/180° = 1/4 of the semi-circle

∴ The area of the sector is 1/4 the area of the semicircle

∵ The area of the semicircle = 1/2 π r²

∵ r = 6 feet

∴ The area of the semicircle = 1/2 π (6)² = 1/2 π (36) = 18 π feet²

∴ Area of the sector = 1/4 (18 π) = 4.5 π feet²

- The small sector HGI has the same central angle of the sector BGC

∴ The area of the sector HGI is 1/4 The area of the small semicircle

∵ The area of the small semicircle = 1/2 π r²

∵ r = 3 feet

∴ The area of the small semicircle = 1/2 π (3)² = 1/2 π (9) = 4.5 π feet²

∴ Area of the sector HGI= 1/4 (4.5 π) = 1.125 π feet²

- The area of the glass in region BCIH is the difference between the

  area of sector BGC and the area of the sector HGI

∴ The area of the glass in region BCIH = 4.5 π - 1.125 π ≅ 11 feet²

Answer:

11 feet to the nearest square foot.

Step-by-step explanation:

The area of sector BCG

= 45/180 * 1/2 π r^2

= 1/4 * 1/2 π r^2

= 1/8 * π * 6^2

= 4.5 π ft^2.

The radius of the inner semicircle is 6 - 3 = 3 feet.

The area of sector HIG = 1/4 * 1/2 π  3^2

= 1.125 π ft^2.

So the area of BCIH

= area of BCG - area HIG

=  4.5  π - 1.125  π

=  3.375  π

= 10.6 square feet.