Question

1. same as one of the problems in wiley the profit (in thousands of dollars) a company makes from selling a certain item depends on the price of the item. the three different forms for the profit at a price of p dollars are: standard form: -2p² + 24p - 54 factored form: -2(p - 3)(p - 9) vertex form: -2(p - 6)² + 18 a) sketch a rough graph of the function, labeling your axes: b) which form is most useful for finding the prices that give a profit of zero dollars? (these are called the break - even prices.) use it to find these prices. form: ____________ prices (with units): __________ c) which form is most useful for finding the profit when the price is zero? use it to find that profit. form: __________ profit when price is zero (with units): __________ what does it mean for profit to be negative? does that make sense? __________ d) the company would like to maximize profits. which form is most useful for finding the price that gives the maximum profit? use it to find the optimal price and the maximum profit. form: __________ optimal price (with units): __________ max profit (with units): ____________

Explanation:

Step1: Analyze graph components

Use factored form for \(x\) - intercepts and vertex form for vertex.

Step2: Find break - even prices

Use zero - product property on factored form.

Step3: Find profit at \(p = 0\)

Substitute \(p = 0\) into standard form.

Step4: Find optimal price and max profit

Use vertex of vertex - form.

Answer:

a)

1. First, note the general form of a quadratic function \(y = ax^{2}+bx + c\). Here \(a=-2\), \(b = 24\), \(c=-54\). The parabola opens downwards since \(a=-2<0\).
2. The \(x\) - intercepts can be found from the factored form \(-2(p - 3)(p - 9)=0\), so \(p = 3\) and \(p = 9\).
3. The vertex can be found from the vertex - form \(-2(p - 6)^{2}+18\), so the vertex is at \((6,18)\). Plot the \(x\) - intercepts at \(p = 3\) and \(p = 9\) and the vertex at \((6,18)\) and draw a parabola opening downwards. Label the \(x\) - axis as "Price (\(p\) in dollars)" and the \(y\) - axis as "Profit (in thousands of dollars)".

b)
Form: Factored form \(-2(p - 3)(p - 9)\)
Prices (with units): Set \(-2(p - 3)(p - 9)=0\). Using the zero - product property, if \(ab = 0\), then \(a = 0\) or \(b = 0\). So \(p-3=0\) gives \(p = 3\) and \(p - 9=0\) gives \(p = 9\). The break - even prices are \(p = 3\) dollars and \(p = 9\) dollars.
c)
Form: Standard form \(-2p^{2}+24p - 54\)
Profit when price is zero (with units): Substitute \(p = 0\) into the standard form: \(y=-2(0)^{2}+24(0)-54=-54\). The profit when the price is zero is \(- 54\) thousand dollars. A negative profit means the company is making a loss. It makes sense because if the company gives the item away (\(p = 0\)), it still has costs associated with production, so it will lose money.
d)
Form: Vertex form \(-2(p - 6)^{2}+18\)
Optimal price (with units): The vertex of a parabola \(y=a(x - h)^{2}+k\) is at \((h,k)\). For the profit function \(y=-2(p - 6)^{2}+18\), the optimal price \(p = 6\) dollars.
Max profit (with units): Substitute \(p = 6\) into the vertex - form. The maximum profit is \(18\) thousand dollars.