Question

The function f(x) = –x2 + 16x – 60 models the daily profit, in dollars, a shop makes for selling candles, where x is the number of candles 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? (5 points)

Part B: Determine the x-intercepts. What do these values mean in the context of the problem? (5 points

please help

Answer 1

The vertex of the profit function f(x) = -x^2 + 16x - 60 is (8, 4), meaning the maximum profit is $4 when 8 candles are sold. The x-intercepts are (6,0) and (10,0), indicating zero profit when either 6 or 10 candles are sold.

Step-by-step explanation:

The function given is a quadratic equation of the form f(x) = ax2 + bx + c, where the constants are a = -1, b = 16, and c = -60. To find the vertex of the parabola, we can use the formula -b/(2a) for the x-coordinate of the vertex.

Calculate the x-coordinate of the vertex: x = -b/(2a) = -16/(2 * -1) = 8.

Substitute x = 8 into the original function to find the y-coordinate: f(8) = -(8)2 + 16(8) - 60 = -64 + 128 - 60 = 4.

The vertex is at the point (8, 4). In the context of the problem, this point represents the maximum daily profit of $4, which occurs when 8 candles are sold.

To determine the x-intercepts, we set f(x) to zero and solve for x, which represents the number of candles sold when the profit is zero.

Set the function equal to zero and solve for x: 0 = -x2 + 16x - 60.

Apply the quadratic formula x = [-b ± sqrt(b2 - 4ac)]/(2a) to find the x-intercepts. In this case, a = -1, b = 16, and c = -60.

Calculate the discriminant, sqrt(162 - 4(-1)(-60)) = sqrt(256 - 240) = sqrt(16) = 4.

Find the two x-intercepts: x = [16 ± 4]/(2 * -1) which yields x = 6 and x = 10.

The x-intercepts (6,0) and (10,0) indicate the number of candles sold at which the profit would be zero, specifically at 6 and at 10 candles sold.

Answer 2

A.

f is a quadratic function, which means it's graph is a parabola.

Notice that the coefficient of
`x^(2)` is negative, so the parabola opens downwards.

the x-coordinate of a parabola is always determined by the formula:
`-(b)/(2a)`

where a is coefficient of the
`x^(2)` term, and b is the coefficient of the x term.

Thus, x-coordinate of the vertex of the graph of f is :


`-(b)/(2a)=-(16)/(2(-1))=8`

the y-coordinate of the vertex is f(8)=-8*8+16*8-60=4.

The vertex is (8, 4).

This means that the maximum daily profit is when exactly 8 candles are sold.

B.

The x-intercepts are the values of x such that f(x)=0,

so to find these values we solve:


`- x^(2) +16x-60=0`



`x^(2)-16x+60=0`

complete the square:


`x^(2)-2*8x+64-64+60=0`


`(x-8)^(2)-4=0`


`(x-8)^(2)= 2^(2)`

so x-8=2 or x-8=-2

the roots are x=10 and x=6, are the roots.

This means that when the shop sells exactly 6 or 10 candles, it makes no profit.