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)

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

Step-by-step explanation:

Let

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

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

we have

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

Complete the square

Rewrite as perfect squares

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

For f(x)=0

solve for x

take square root both sides

so

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

substitute the value of x

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)

the largest number of the four options is 1.8 x 10-2

you have it in scientific notation.

1.80 x 10-6   = 0.00000180

1.80 x 10-4   = 0.000180 (bigger, or "less small", than the former one)

so

1.8 x 10-2       = 0.0180. as you see. the "less small", or the biggest one of them all.

as you have negative exponent, it means you are dividing 1.8 by 100 (i.e. moving the dot towards left, two times).

if you had positive exponent for the scientific notation, power of tenths, you would be multiplying times 100 (moving the dot towards right) as below:

1.8 x (10)2 = 180

so. if you have negative exponent in scientific notation, move the dot toward left. if it is positive, toward right as many times the exponent tells you.

111

step-by-step explanation: