Question

The manager of Fence Me In is trying to determine the best selling price for a particular type of gate latch. The function

p(s)=4s2+400s8400p(s) = -4s^2 + 400s – 8400

models the yearly profit the company will make from the latches when the selling price is s dollars. (a). Write a quadratic equation that can be used to determine the selling price that would result in a yearly profit of 1600.(b).Writethequadraticequationinstandardformsothatthecoefficientof1600. (b). Write the quadratic equation in standard form so that the coefficient of s2s^2$ is 1. (c). Solve the quadratic equation by factoring, and interpret the solution(s). (d). Explain how you could check your answer to part c.

Solution

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(a)\textbf{(a)}

Because pp represents the yearly profit, then we substitute p=1600p=1600 to obtain the equation:

1600=4s2+400s84001600=-4s^2+400s-8400

or

4s2400s+10000=0\color{#c34632}{4s^2-400s+10000=0}

(b)\textbf{(b)}

We divide both sides of th equation by 4 to obtain:

s2100s+2500=0\color{#c34632}{s^2-100s+2500=0}

(c)\textbf{(c)}

Factor the left side:

(s50)(s50)=0(s-50)(s-50)=0

By Zero Product Property,

s=50\color{#c34632}{s=50}

This means that in order to have a yearly profit of $1600, the selling price of the latches shpuld be $50.\ \ \textbf{(d)}\

We can substitutes=50s=50 to the yearly profit function and check if the profit is $1600:

p(50)=4(50)2+400(50)8400=1600p(50)=-4(50)^2+400(50)-8400=1600\hspace{1mm}\checkmark

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