Question

9.6 graph quadratic using properties (homework)
score: 9/20 answered: 9/20
question 14
a store manager determines that the revenue from shoes, when the price for a pair of shoes is t dollars,
will be ( h(t)=-t^{2}+36 t ) dollars.
what price should be charged to maximize revenue? dollars
what will the revenue be at this price? dollars

Explanation:

Step1: Find the price \( t \) that maximizes revenue

For a quadratic function \( y = ax^{2}+bx + c \), the vertex (which gives the maximum or minimum value) has its \( x \)-coordinate at \( t=-\frac{b}{2a} \).
In the function \( h(t)=-t^{2}+36t \), \( a=- 1\) and \( b = 36 \).
Using the formula \( t=-\frac{b}{2a} \), we substitute the values:
\( t=-\frac{36}{2\times(-1)}=\frac{-36}{-2}=18 \)

Step2: Find the maximum revenue

Substitute \( t = 18 \) into the function \( h(t)=-t^{2}+36t \).
\( h(18)=-(18)^{2}+36\times18 \)
First, calculate \( (18)^{2}=324 \) and \( 36\times18 = 648 \)
\( h(18)=-324 + 648=324 \)

Answer:

The price to maximize revenue is \( 18 \) dollars. The revenue at this price is \( 324 \) dollars.