Question

The function f(x) = −x2 + 44x − 384 models the hourly profit, in dollars, a shop makes for selling coffee, where x is the number of cups of coffee sold, and f(x) is the amount of profit. Part A: Determine the vertex. What does this calculation mean in the context of the problem? (4 points) Part B: Determine the x-intercepts. What do these values mean in the context of the problem? (4 points) Part C: Determine the y-intercept. What does this value mean in the context of the problem? (2 points) (10 points)

Answer 1

Part A) The vertex is the point (22,100) see the explanation

Part B) The x-intercepts are the points (12,0) and (32,0 see the explanation

Part C) The y-intercept is the point (0,-384) see the explanation

Explanation:

Let

x ----> the number of cups of coffee sold

f(x) ---> the amount of profit

we have

`f(x)=-x^(2) +44x-384`

This is a vertical parabola open downward (the leading coefficient is negative)

The vertex represent a maximum

Part A) Determine the vertex. What does this calculation mean in the context of the problem?

Convert the quadratic equation in vertex form

Factor -1

`f(x)=-(x^(2)-44x)-384`

Complete the square

`f(x)=-(x^(2)-44x+22^2)-384+22^2`

`f(x)=-(x^(2)-44x+484)+100`

Rewrite as perfect squares

`f(x)=-(x-22)^(2)+100`

The vertex is the point (22,100)

That means ----> The maximum profit of $100 is when the number of cups of coffee sold is 22

Part B) Determine the x-intercepts. What do these values mean in the context of the problem?

we know that

The x-intercepts are the values of x when the value of the function is equal to zero

so

we have

`f(x)=-(x-22)^(2)+100`

For f(x)=0

`0=-(x-22)^(2)+100`

solve for x

`(x-22)^(2)=100`

take square root both sides

`(x-22)=\pm10`

`x=22\pm10`

so

`x=22+10=32`

`x=22-10=12`

The x-intercepts are the points (12,0) and (32,0)

That means -----> When the number of cups of coffee sold is 12 or 32 the profit is equal to zero

Part C) Determine the y-intercept. What does this value mean in the context of the problem?

we know that

The y-intercept is the value of the function when the value of x is equal to zero

so

For x=0

`f(x)=-x^(2) +44x-384`

substitute the value of x

`f(x)=-(0)^(2) +44(0)-384`

`f(x)=-384`

The y-intercept is the point (0,-384)

That means ----> When the number of cups of coffee sold is zero the profit is negative -$384 (the cost is greater than the revenue)