Quadratic Equation Application

Question

Textbook exercise I get stuck

"Amy is going to hold a concert at a stadium. The stadium can accommodate 12 000 people. If the price for each ticket is \$160, all the tickets will be sold. For every increase of \$1 in the ticket price, the number of tickets sold will decrease by 50. Let \$p be the price of each ticket and q be the number if tickets sold

(a) express $q$ in terms of $p$, where $p \geq 160$."

Please help!

$q$ will be a linear function of $p$. You're given one $(p,q)$ pair; you just told me another. So can you find $q$ in general now? — Ian, Dec 01 '15 at 14:51
'(b) Find the price of each ticket in order to get the maximum income' That is the part (b) of the question — user256670, Dec 01 '15 at 14:54
Do you understand why $q$ is a linear function of $p$? If so, you know $q(160)=12000$ and $q(161)=11950$, so you should be able to compute $q(p)$ in general. — Ian, Dec 01 '15 at 14:54

Frédéric · Answer 1 · 2015-12-01T15:34:12.660

The starting number of tickets sold is $12000\,\text{tickets}$. For each dollar above $160\,\$$, 50 fewer tickets are sold. The number of dollars above $160\,\$$ is \begin{equation} p-160\,\$\;. \end{equation} The number of tickets sold is therefore \begin{equation} q = 12000\,\text{tickets} - \left(50\,\frac{\text{tickets}}{\$}\right)(p - 160\,\$)\;. \end{equation}

Make sure you understand every part of the previous expression before moving on to part (b).

For part (b), you must use the total revenue from ticket sales: \begin{align} \text{Total revenue} &= \text{tickets sold}\times\text{price of tickets}\\ &=q\times p\\ &=\left(12000\,\text{tickets} - \left(50\,\frac{\text{tickets}}{\$}\right)(p - 160\,\$)\right)p\\ &\vdots \end{align} Now you can find the vertex or max point of the quadratic function.

I answered the question asked. One question, one answer. Ask another question, get another answer. I will add a small part to my answer. — Frédéric, Dec 01 '15 at 15:30

Quadratic Equation Application

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