Q. 75 - Precalculus Enhanced with Graphing Utilities

Question

Theater Revenues 

A Broadway theater has $500$ seats, divided into orchestra, main, and balcony seating. Orchestra seats sell for $\backslash (50$, main seats for $\backslash )35$, and balcony seats for $\backslash (25$. If all the seats are sold, the gross revenue to the theater is $\backslash )17,100$. If all the main and balcony seats are sold, but only half the orchestra seats are sold, the gross revenue is $\$14,600$. How many are there of each kind of seat? 

Step-by-Step Solution

Answer

There are $100$ orchestra seats, $210$ main seats, and $190$ balcony seats.

Step 1. Given Information

Total seats $=500$

Cost of Orchestra seat $=\$50$

Cost of Main seat $=\$35$

Cost of Balcony seat $=\$25$

Gross Revenue when all seats are sold $=\$17100$

Gross Revenue when all the main and balcony seats are sold, but only half the orchestra seats are sold $=\$14600$

Step 2. Using substitution and elimination method

Let $x$ denote the number of Orchestra seats, $y$denote main seats and $z$ denote balcony seats.

The required equations are:

$$\begin{gathered} x+y+z=500.......\left(i\right) \\ 50x+35y+25z=17100........\left(ii\right) \\ 25x+35y+25z=14600........\left(iii\right) \end{gathered}$$

Subtracting equation $\left(iii\right)$ from equation $\left(ii\right)$ we get,

$$\begin{gathered} 25x=2500 \\ \Rightarrow x=\frac{2500}{25} \\ \Rightarrow x=100 \end{gathered}$$

Substituting the value of $x$ in equations $\left(i\right),\left(ii\right)$ we get:

$$\begin{gathered} 20+y+z=500 \\ \Rightarrow y+z=500-20 \\ \Rightarrow y+z=480.......\left(iv\right) \\ 50\left(100\right)+35y+25z=17100 \\ \Rightarrow 5000+35y+25z=17100 \\ \Rightarrow 35y+25z=17100-5000 \\ \Rightarrow 35y+25z=12100.........\left(v\right) \end{gathered}$$

Multiplying equation $\left(iv\right)$ by $25$ we get,

$$\begin{gathered} 25y+25z=25\times 400 \\ \Rightarrow 25y+25z=10000........\left(vi\right) \end{gathered}$$

Subtracting equation $\left(vi\right)$ from $\left(v\right)$ we get,

$$\begin{gathered} 10y=2100 \\ \Rightarrow y=\frac{2100}{10} \\ \Rightarrow y=210 \end{gathered}$$

From equation $\left(vi\right)$ we get,

$$\begin{gathered} 25\left(210\right)+25z=10000 \\ \Rightarrow 5250+25z=10000 \\ \Rightarrow 25z=10000-5250 \\ \Rightarrow 25z=4750 \\ \Rightarrow z=\frac{4750}{25} \\ \Rightarrow z=190 \end{gathered}$$