## Related questions with answers

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

$p(s) = -4s^2 + 400s – 8400$

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 $1600. (b). Write the quadratic equation in standard form so that the coefficient of$ $s^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

Verified$\textbf{(a)}$

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

$1600=-4s^2+400s-8400$

or

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

$\textbf{(b)}$

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

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

$\textbf{(c)}$

Factor the left side:

$(s-50)(s-50)=0$

By Zero Product Property,

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

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 substitute$s=50$ to the yearly profit function and check if the profit is $1600:

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

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