Question

The total revenue R earned per day (in dollars) from a pet-sitting service is given by R(p) = -12p² + 174p, where p is the price charged per pet (in dollars). (a) Find the revenue earned for each price per pet given below. R( 6) 612 Excellent job! R( 7) 630 You're right! R( 9) 594 That's it! (b) Find the unit price that yields a maximum revenue. 5 Xper pet What is the maximum revenue? 630.25 X​

Answer 1

Check the picture below.

so that picture is a parabolic path for an object, hmmm so the quadratic equation for R(p) is pretty much about the same, so its peak is at the vertex, so let's find its vertex.

`\textit{vertex of a vertical parabola, using coefficients} \\\\ R(p)=\stackrel{\stackrel{a}{\downarrow }}{-12}p^2\stackrel{\stackrel{b}{\downarrow }}{+174}p\stackrel{\stackrel{c}{\downarrow }}{+0} \qquad \qquad \left(-\cfrac{ b}{2 a}~~~~ ,~~~~ c-\cfrac{ b^2}{4 a}\right)`

`\left(-\cfrac{ 174}{2(-12)}~~~~ ,~~~~ 0-\cfrac{ (174)^2}{4(-12)}\right) \implies\left( - \cfrac{ 174 }{ -24 }~~,~~0 - \cfrac{ 30276 }{ -48 } \right) \\\\\\ \left( \cfrac{ 174 }{ 24 }~~,~~0 + \cfrac{ 30276 }{ 48 } \right)\implies \left( \cfrac{ 174 }{ 24 }~~,~~0 + 630.75 \right)\implies (~\stackrel{ p }{7.25}~~,~~ \stackrel{ R(p) }{630.75}~)`

Answer 2

$630.25.

Explanation:

AI-generated answer

To find the revenue earned for each price per pet, we substitute the given price values into the revenue function R(p) = -12p² + 174p.

(a) Revenue earned for each price per pet:

- R(6): R(6) = -12(6)² + 174(6) = 612 dollars

- R(7): R(7) = -12(7)² + 174(7) = 630 dollars

- R(9): R(9) = -12(9)² + 174(9) = 594 dollars

(b) To find the unit price that yields a maximum revenue, we can use calculus. We need to find the vertex of the parabola represented by the revenue function.

The revenue function R(p) = -12p² + 174p is a quadratic function in the form of ax² + bx + c. The vertex of a quadratic function with this form is given by the formula:

p = -b / (2a)

In this case, a = -12 and b = 174. Let's substitute these values to find the unit price that yields a maximum revenue:

p = -174 / (2(-12)) = 7.25

So, the unit price that yields a maximum revenue is $7.25 per pet.

To find the maximum revenue, we substitute this unit price into the revenue function:

R(7.25) = -12(7.25)² + 174(7.25) = 630.25 dollars