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A manager estimates a band's profit p for a concert by using the function p(t)=-200t^2+2500t-c, where t is the price per ticket and c is the band's operation cost. The table shows the band's operating cost at three different concert locations. What range of ticket prices should the band charge at each location in order to make a profit of at least $1000 at each concert? Band's Costs Location Operating Cost Freemont Park$900 Saltillo Plaza $1500 Riverside Walk$2500

Question
A manager estimates a band's profit p for a concert by using the function p(t)=-200t^2+2500t-c, where t is the price per ticket and c is the band's operation cost. The table shows the band's operating cost at three different concert locations. What range of ticket prices should the band charge at each location in order to make a profit of at least $1000 at each concert? Band's Costs Location Operating Cost Freemont Park$900
Saltillo Plaza $1500 Riverside Walk$2500

Answers (1)

2020-12-03
freemont park: c=900
p(t) = 1000 subsitute variables into equation
$$\displaystyle{p}{\left({t}\right)}=-{200}{\left({t}^{{2}}\right)}+{2500}{t}-{c}$$
$$\displaystyle{\left({1000}\right)}=-{200}{\left({t}^{{2}}\right)}+{2500}{t}-{\left({900}\right)}$$
$$\displaystyle{0}=-{200}{\left({t}^{{2}}\right)}+{2500}{t}-{1900}$$
$$\displaystyle{200}{t}^{{2}}-{2500}{t}+{1900}={0}$$
a = 200, b = -2500 c = 1900 quadratic equation
$$\displaystyle{t}=\frac{{{2500}\pm\sqrt{{{2500}^{{2}}-{4}{\left({200}\right)}{\left({1900}\right)}}}}}{{2}}{\left({200}\right)}$$
$$\displaystyle{t}=\frac{{{2500}\pm\sqrt{{{4730000}}}}}{{400}}$$
$$\displaystyle{t}=\frac{{25}}{{4}}+\frac{\sqrt{{{473}}}}{{4}}$$
$$\displaystyle{t}=\frac{{25}}{{4}}-\frac{\sqrt{{{473}}}}{{4}}$$
$$\displaystyle{t}={6.25}+{5.43714079273}=\{11.69}$$ or
$$\displaystyle{t}={6.25}-{5.43714079273}=\{0.81}$$
saltillo plaza: c = 1500
p(t) = 1000 subsitute variables into equation
$$\displaystyle{p}{\left({t}\right)}=-{200}{\left({t}^{{2}}\right)}+{2500}{t}-{c}$$
$$\displaystyle{\left({1000}\right)}=-{200}{\left({t}^{{2}}\right)}+{2500}{t}-{\left({1500}\right)}$$
$$\displaystyle{0}=-{200}{\left({t}^{{2}}\right)}+{2500}{t}-{2500}$$
$$\displaystyle{200}{t}^{{2}}-{2500}{t}+{2500}={0}$$
a = 200, b = -2500 c = 2500 quadratic equation
$$\displaystyle{t}=\frac{{{2500}\pm\sqrt{{{2500}^{{2}}-{4}{\left({200}\right)}{\left({2500}\right)}}}}}{{2}}{\left({200}\right)}$$
$$\displaystyle{t}=\frac{{{2500}\pm\sqrt{{{4250000}}}}}{{400}}$$
$$\displaystyle{t}=\frac{{25}}{{4}}+{\left(\frac{{5}}{{4}}\right)}\sqrt{{{17}}}=\{11.40}$$ or
$$\displaystyle{t}=\frac{{25}}{{4}}-{\left(\frac{{5}}{{4}}\right)}\sqrt{{{17}}}=\{1.09}$$
riverside walk: c = 2500
p(t) = 1000 subsitute variables into equation
$$\displaystyle{p}{\left({t}\right)}=-{200}{\left({t}^{{2}}\right)}+{2500}{t}-{c}$$
$$\displaystyle{\left({1000}\right)}=-{200}{\left({t}^{{2}}\right)}+{2500}{t}-{\left({2500}\right)}$$
$$\displaystyle{0}=-{200}{\left({t}^{{2}}\right)}+{2500}{t}-{3500}$$
$$\displaystyle{200}{t}^{{2}}-{2500}{t}+{3500}={0}$$
a = 200, b = -2500 c = 3500 quadratic equation
$$\displaystyle{t}=\frac{{{2500}\pm\sqrt{{{2500}^{{2}}-{4}{\left({200}\right)}{\left({3500}\right)}}}}}{{2}}{\left({200}\right)}$$
$$\displaystyle{t}=\frac{{{2500}\pm\sqrt{{{3450000}}}}}{{400}}$$
t = $10.89 or t =$1.61

Relevant Questions

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The table below shows the number of people for three different race groups who were shot by police that were either armed or unarmed. These values are very close to the exact numbers. They have been changed slightly for each student to get a unique problem.
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1
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Suppose the manufacturer of widgets has developed the following table showing the highest price p, in dollars, of a widget at which N widgets can be sold.
$$\begin{array}{|c|c|} \hline Number\ N & Price\ p\\ \hline 200 & 53.00\\ \hline 250 & 52.50\\\hline 300 & 52.00\\ \hline 350 & 51.50\\ \hline \end{array}$$
(a) Find a formula for p in terms of N modeling the data in the table.
$$\displaystyle{p}=$$
(b) Use a formula to express the total monthly revenue R, in dollars, of this manufacturer in a month as a function of the number N of widgets produced in a month.
$$\displaystyle{R}=$$
Is R a linear function of N?
(c) On the basis of the tables in this exercise and using cost, $$\displaystyle{C}={35}{N}+{900}$$, use a formula to express the monthly profit P, in dollars, of this manufacturer as a function of the number of widgets produced in a month.
$$\displaystyle{P}=$$
(d) Is P a linear function of N?
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