Question

The owner of a video store has determined that the profits p of the store are approximately given by p(x) = -x^2 + 110x + 54, where x is the number of videos rented daily. Find the maximum profit to the nearest dollar.

Answer 1

To find the maximum profit for the video store, calculate the vertex of the quadratic profit function by using the formula =x = -b/(2a)=, which gives 55 videos rented daily. Substitute this value back into the profit function to get a maximum profit of $3079.

Step-by-step explanation:

The owner of a video store has determined that the profits p of the store are approximately given by p(x) = -x^2 + 110x + 54, where x is the number of videos rented daily. To find the maximum profit, we need to find the vertex of the parabolic function, since the coefficient of x^2 is negative, indicating a downward opening parabola, and hence the vertex represents the maximum point.

To find the vertex, we use the formula x = -b/(2a), where a and b are the coefficients from the quadratic equation ax^2 + bx + c. Substituting the values from the profit function, we get:

• a = -1
• b = 110

Thus, the number of videos rented for maximum profit is:

x = -110/(2 * -1) = 55

Now, to find the maximum profit, we substitute x = 55 back into the profit function:

p(55) = -(55)^2 + 110*(55) + 54

p(55) = -3025 + 6050 + 54

p(55) = 3025 + 54

p(55) = 3079

Therefore, the maximum profit to the nearest dollar is $3079.