Mathematics : asked on 8ed12c498ffe6ce43a93f916056180ed

26.09.2021

Calculate the unit rate for each of the following. Round to two decimal places and include units.

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22.01.2023, solved by verified expert

Rajat Thapa

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Mathematics, DAV Post Graduate College

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Answer:

Answer written below.

Step-by-step explanation:

a.) 200 miles in 3 hours

So, unit rate = Total Distance/Time taken = 200/3

= 66.67 miles per hour

(b.) 36 inches to 3 feet

So, unit rate = 36/3

= 12 inches per ft

(c.) $21.50 for 4 gallons

So, unit rate = Total amount/Time volume = 21.50/4

= $5.38 per gallon

d.) $2.08 for 6 ounces

So, unit rate = $2.08/6

= $0.35 per ounce

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Mathematics

A model rocket is launched directly upward at a speed of 20 meters per second from the ground. The function f(t)=−4.9t2+20t, models the relationship between the height of the rocket and the time after launch, t, in seconds. When, in seconds after launch, will the rocket reach its highest point? Round to two decimal places.

Step-by-step answer

P Answered by Master

The rocket will reach its highest point about 2.04 seconds.

Step-by-step explanation:

The function:

f(t)=-4.9t^2+20t

Models the relationship between the height of the rocket and the time after launch t in seconds.

Since this is a quadratic function, the rocket will reach its highest point at its vertex. The vertex of a quadratic is given by:

$\displaystyle \Big(-\frac{b}{2a}, f\Big(-\frac{b}{2a}\Big)\Big)$

In this case, a = -4.9, b = 20, and c = 0. Thus:

$\displaystyle t=\frac{-(20)}{2(-4.9)}=\frac{10}{4.9}\approx 2.04$

The rocket reaches its maximum height after 10/4.9 or about 2.04 seconds.

Further Notes:

Then the maximum height of the rocket will be f(10/4.9):

$\displaystyle f\Big(\frac{10}{4.9}\Big)=-4.9\Big(\frac{10}{4.9}\Big)^2+20\Big(\frac{10}{4.9}\Big)\approx20.41\text{ feet}$

Mathematics

Step-by-step answer

P Answered by Master

The rocket will reach its highest point about 2.04 seconds.

Step-by-step explanation:

The function:

f(t)=-4.9t^2+20t

Models the relationship between the height of the rocket and the time after launch t in seconds.

Since this is a quadratic function, the rocket will reach its highest point at its vertex. The vertex of a quadratic is given by:

$\displaystyle \Big(-\frac{b}{2a}, f\Big(-\frac{b}{2a}\Big)\Big)$

In this case, a = -4.9, b = 20, and c = 0. Thus:

$\displaystyle t=\frac{-(20)}{2(-4.9)}=\frac{10}{4.9}\approx 2.04$

The rocket reaches its maximum height after 10/4.9 or about 2.04 seconds.

Further Notes:

Then the maximum height of the rocket will be f(10/4.9):

$\displaystyle f\Big(\frac{10}{4.9}\Big)=-4.9\Big(\frac{10}{4.9}\Big)^2+20\Big(\frac{10}{4.9}\Big)\approx20.41\text{ feet}$

Mathematics

A coral reef is 6 miles due north of its closest point along a straight shoreline. A visitor is staying in a tent that is 9 miles east of that point. The visitor is planning to go from the tent to the coral reef. Suppose the visitor runs at a rate of 4 mph and swims at a rate of 2 mph. How far should the visitor run to minimize the time it takes to reach the coral reef? Round to two decimal places.

Step-by-step answer

P Answered by PhD

5.54 miles

Explanation:

1. Assume she runs up to a point that is x miles to the east of the coral reef:

distance running = $9-x\text{ }miles$ distance swimming = $\sqrt{x^2+6^2}=\sqrt{x^2+36}\text{ }miles$

2. Times:

time= distance / speed

time running = $\dfrac{9-x}{4}$ time swimming = $\dfrac{\sqrt{x^2+36} }{2}$

3. Total time

time running + time swimming:

$T=\dfrac{9-x}{4}+\dfrac{\sqrt{x^2+36} }{2}$

4. Minimize time

The minimum time is when the derivative of the function of time is zero.

Find the derivative of T(x), using rule of the sum, the derivative of the sum is the sum of the derivatives, and the chain rule:

$\dfrac{dT}{dx}=\dfrac{-1}{4}+\dfrac{1}{2}\cdot \dfrac{1}{2}\dfrac{2x}{\sqrt{x^2+36}}=0$

Solve for x:

$\dfrac{2x}{\sqrt{x^2+36}}=1\\ \\ \\ 2x=\sqrt{x^2+36\\ } \\ \\ (2x)^2=x^2+36\\ \\ \\ 4x^2=x^2+36\\ \\ \\ 3x^2=36\\ \\ \\ x^2=12\\ \\ \\ x=2\sqrt{3}$

5. Distance the visitor should run to minimize the time

Hence, the visitor should run $9-2\sqrt{3}\approx 5.54$ miles.

Mathematics

Asap - a right triangle has an angle of 38 degrees and an opposite leg of 12 yards. solve the right triangle. label each of your solutions. round to two decimals.

Step-by-step answer

P Answered by PhD

ANSWER

See below

EXPLANATION

To solve the triangle means to find all sides and angles.

The right triangle has an angle of 38 degrees and an opposite leg of 12 yards

The third angle is 90-38=52°

We use the sine ratio to find the hypotenuse, h.

$\sin(38 \degree) = \frac{opposite}{hypotenuse}$

$\sin(38 \degree) = \frac{12}{h}$

$h = \frac{12}{ \sin(38) }$

h = 19.49

The hypotenuse is 19.49 yards to the nearest hundredth.

We can find the adjacent leg using the cosine ratio.

$\cos(38) = \frac{adjacent}{hypotenuse}$

$\cos(38) = \frac{a}{19.49}$

$a = 19.49 \cos(39)$

a = 15.36

to the nearest hundredth.

Mathematics

Step-by-step answer

P Answered by PhD

5.54 miles

Explanation:

1. Assume she runs up to a point that is x miles to the east of the coral reef:

distance running = $9-x\text{ }miles$ distance swimming = $\sqrt{x^2+6^2}=\sqrt{x^2+36}\text{ }miles$

2. Times:

time= distance / speed

time running = $\dfrac{9-x}{4}$ time swimming = $\dfrac{\sqrt{x^2+36} }{2}$

3. Total time

time running + time swimming:

$T=\dfrac{9-x}{4}+\dfrac{\sqrt{x^2+36} }{2}$

4. Minimize time

The minimum time is when the derivative of the function of time is zero.

Find the derivative of T(x), using rule of the sum, the derivative of the sum is the sum of the derivatives, and the chain rule:

$\dfrac{dT}{dx}=\dfrac{-1}{4}+\dfrac{1}{2}\cdot \dfrac{1}{2}\dfrac{2x}{\sqrt{x^2+36}}=0$

Solve for x:

$\dfrac{2x}{\sqrt{x^2+36}}=1\\ \\ \\ 2x=\sqrt{x^2+36\\ } \\ \\ (2x)^2=x^2+36\\ \\ \\ 4x^2=x^2+36\\ \\ \\ 3x^2=36\\ \\ \\ x^2=12\\ \\ \\ x=2\sqrt{3}$

5. Distance the visitor should run to minimize the time

Hence, the visitor should run $9-2\sqrt{3}\approx 5.54$ miles.

Mathematics

Asap - a right triangle has an angle of 38 degrees and an opposite leg of 12 yards. solve the right triangle. label each of your solutions. round to two decimals.

Step-by-step answer

P Answered by PhD

ANSWER

See below

EXPLANATION

To solve the triangle means to find all sides and angles.

The right triangle has an angle of 38 degrees and an opposite leg of 12 yards

The third angle is 90-38=52°

We use the sine ratio to find the hypotenuse, h.

$\sin(38 \degree) = \frac{opposite}{hypotenuse}$

$\sin(38 \degree) = \frac{12}{h}$

$h = \frac{12}{ \sin(38) }$

h = 19.49

The hypotenuse is 19.49 yards to the nearest hundredth.

We can find the adjacent leg using the cosine ratio.

$\cos(38) = \frac{adjacent}{hypotenuse}$

$\cos(38) = \frac{a}{19.49}$

$a = 19.49 \cos(39)$

a = 15.36

to the nearest hundredth.

Business

The following financial information was obtained from the year ended 2018 income statements for Kringle Automotive and Leihman Automotive (Click the icon to view the financial information.) Requirements 1. Compute the times-interest-earned ratio for each company. Round to two decimals 2. Which company was better able to cover its interest expense? Requirement 1. Compute the times-interest-earned ratio for each company. Round to two decimals. Begin by showing the formula for the times-interest-earned ratio. Times-interest-earned ratio Now calculate

Step-by-step answer

P Answered by Specialist

times interest ratio = EBIT / interest expense

Kringle: TIE ratio = ($40,870 - $10,300 - $350) / $350 = 86.34

Leihman: TIE ratio = ($46,320 - $12,080 - $2,000) / $2,000 = 16.12

Kringle was better able to cover its interest expenses since its TIE ratio is much higher than Leihman's (more than 5 times higher). This means that it is much easier for Kringle to pay off the interests on its debt.

The following financial information was obtained from the year ended 2018 income statements for Krin

Mathematics

The value, V (t), in dollars, of a stock t months after it is purchased is modeled by the following equation. V(t)= 42(1-e ^-1.5t) +36 a) V (1) =$. (Round to two decimal places as needed) V (12) =$.(Round to two decimal places as needed) b)Find V (t)=.(Simply your answer. Use intergers or decimal for any numbers in the expression. Round to one decimal place as needed) c) After how many months will the value of the stock first reach $75?( Do not round until the final answer. Then round to one decimal place as needed) d)lim V (t)= $ t-- [infinity]

Step-by-step answer

P Answered by Master

Step-by-step explanation: $V(t)= 42(1-e ^{1.5t} ) +36$ is given value in dollars of a stock in t months after it is purchased.

a) Substitute 1 for t

$V(1) = 42(1-e^{-1.5} )+36 \\=67.852\\=67.85$

V(12) = $V(1) = 42(1-e^{-1.5*12} )+36 \\=77.00$

b) Find derivative for V

$V'(t) = 42(-1.5) e^{-1.5t} )\\\\=-63e^{-1.5t}$

c) When V(t) = 75

$V(t)=75= 42(1-e ^{1.5t} ) +36\\1-e ^{1.5t=0.9286\\=-2.639$

d) As t tends to infinity, exponent being in negative t tends to 0

So V tends to 42(1-0)+36 = 78

Mathematics

Step-by-step answer

P Answered by Specialist

Step-by-step explanation: $V(t)= 42(1-e ^{1.5t} ) +36$ is given value in dollars of a stock in t months after it is purchased.

a) Substitute 1 for t

$V(1) = 42(1-e^{-1.5} )+36 \\=67.852\\=67.85$

V(12) = $V(1) = 42(1-e^{-1.5*12} )+36 \\=77.00$

b) Find derivative for V

$V'(t) = 42(-1.5) e^{-1.5t} )\\\\=-63e^{-1.5t}$

c) When V(t) = 75

$V(t)=75= 42(1-e ^{1.5t} ) +36\\1-e ^{1.5t=0.9286\\=-2.639$

d) As t tends to infinity, exponent being in negative t tends to 0

So V tends to 42(1-0)+36 = 78

Business

Blossom taxi service uses the units-of-activity method in computing depreciation on its taxicabs. each cab is expected to be driven 167,000 miles. taxi 10 cost $33,600 and is expected to have a salvage value of $200. taxi 10 was driven 32,000 miles in 2021 and 32,700 miles in 2022. determine the depreciation cost. (round answer to 2 decimal places, e.g. 1.25.) depreciable cost $enter the depreciation per unit rounded to two decimal places in dollars per unit etextbook and media compute the depreciation for each year. 2021 2022 depreciation expense

Step-by-step answer

P Answered by PhD

$0.20 per miles, $6,400, $6,540

Explanation:

The calculation of the depreciation per unit and its expense method is presented below:

= (Original cost - residual value) ÷ (estimate driven miles)

= ($33,600 - $200) ÷ (167,000 miles)

= ($33,400) ÷ (167,000 miles)

= $0.20 per miles

Now for the 2021 year, it would be

= Miles driven in 2021 year × depreciation per miles

= 32,000 miles × $0.20

= $6,400

And, in 2022, it would be

= Miles driven in 2022 year × depreciation per miles

= 32,700 miles × $0.20

= $6,540

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Calculate the unit rate for each of the following. Round to two decimal places and include units.

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