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kenny6666 [7]
3 years ago
13

A car rental costs $50 for the first day. Additional days cost $35 per day, unless the car is rented for 7 days or more, in whic

h case there is a 10% discount on the daily rate. Identify the expression which represents the cost of renting a car if the car has been rented for more than a week.
Mathematics
1 answer:
son4ous [18]3 years ago
6 0

Answer:

\$50+\$31.5x

Step-by-step explanation:

Let

x------> the number of days

y----> the cost of renting a car

we know that

For x

y=\$50+\$35x

For x\geq 7\ days

The rate is equal to

0.90*\$35=\$31.5

so

y=\$50+\$31.5x

In this problem. the car has been rented for more than a week

therefore

x> 7\ days

The cost of renting a car is equal to

y=\$50+\$31.5x

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For the remaining integral, let t=\tan\dfrac x2. Then

\sin x=\sin\left(2\times\dfrac x2\right)=2\sin\dfrac x2\cos\dfrac x2=\dfrac{2t}{1+t^2}
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and

\mathrm dt=\dfrac12\sec^2\dfrac x2\,\mathrm dx\implies \mathrm dx=2\cos^2\dfrac x2\,\mathrm dt=\dfrac2{1+t^2}\,\mathrm dt

Now the integral is

\displaystyle\int\mathrm dx+2\int\frac{\dfrac{2t}{1+t^2}+3}{\dfrac{1-t^2}{1+t^2}+\dfrac{2t}{1+t^2}+1}\times\frac2{1+t^2}\,\mathrm dt

The first integral is trivial, so we'll focus on the latter one. You have

\displaystyle2\int\frac{2t+3(1+t^2)}{(1-t^2+2t+1+t^2)(1+t^2)}\,\mathrm dt=2\int\frac{3t^2+2t+3}{(1+t)(1+t^2)}\,\mathrm dt

Decompose the integrand into partial fractions:

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so you have

\displaystyle2\int\frac{3t^2+2t+3}{(1+t)(1+t^2)}\,\mathrm dt=4\int\frac{\mathrm dt}{1+t}+2\int\frac{\mathrm dt}{1+t^2}+\int\frac{2t}{1+t^2}\,\mathrm dt

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To try to get the terms to match up with the available answers, let's add and subtract \ln\left|1+\tan\dfrac x2\right| to get

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