question 7 (1 point) a car rental company rents 300 cars at a cost of $30 per car on a weekend. for every $3…

question 7 (1 point) a car rental company rents 300 cars at a cost of $30 per car on a weekend. for every $3 increase in cost, the company rents 10 fewer cars. determine the maximum revenue. a) $12 000 b) $18 000 c) $15 000 d) $9000

question 7 (1 point) a car rental company rents 300 cars at a cost of $30 per car on a weekend. for every $3 increase in cost, the company rents 10 fewer cars. determine the maximum revenue. a) $12 000 b) $18 000 c) $15 000 d) $9000

Answer

Explanation:

Step1: Define variables

Let $x$ be the number of $3 - increases in cost. The cost per car is $C = 30+3x$, and the number of cars rented is $N=300 - 10x$.

Step2: Form the revenue function

Revenue $R = C\times N=(30 + 3x)(300-10x)$. Expand the function: [ \begin{align*} R&=30\times300-30\times10x+3x\times300-3x\times10x\ &=9000-300x + 900x-30x^{2}\ &=- 30x^{2}+600x + 9000 \end{align*} ]

Step3: Find the vertex of the quadratic function

For a quadratic function $y = ax^{2}+bx + c$ ($a=-30$, $b = 600$, $c = 9000$), the $x$ - coordinate of the vertex is $x=-\frac{b}{2a}$. [x=-\frac{600}{2\times(-30)}=10]

Step4: Calculate the maximum revenue

Substitute $x = 10$ into the revenue function $R=-30x^{2}+600x + 9000$. [ \begin{align*} R&=-30\times10^{2}+600\times10 + 9000\ &=-3000+6000 + 9000\ &=12000 \end{align*} ]

Answer:

A. $12000$