# A word problem

• May 14th 2009, 06:59 AM
dizizviet
A word problem
I've always had a problem with word problems...

1)A baseball team is trying to determine what price to charge for tickets. At a price of \$10 per ticket, it averages 45,000 people per game. For every increase of \$1, it loses 5,000 people. Everyone person at the game spends an average of \$5 on concessions. What price per ticket should be charged in order to maximize revenue?

2) The volume of a rectangular box with a square base remains constant at 400 cm^3 as the area of the base increases as a rate of 15cm^3/sec. Find the rate at which the height of the box is decreasing when each side of the base is 19 cm long. (don't round answer)
• May 14th 2009, 10:46 AM
Soroban
Hello, dizizviet!

Here's the first one . . .

Quote:

1) A baseball team is trying to determine what price to charge for tickets.
At a price of \$10 per ticket, it averages 45,000 people per game.
For every increase of \$1, it loses 5,000 people.
Everyone person at the game spends an average of \$5 on concessions.
What price per ticket should be charged in order to maximize revenue?

Let \$\displaystyle x\$ = number of \$1 increases in the price of ticket.

They will charge \$\displaystyle (10 + x)\$ dollars per ticket.

For each \$\displaystyle x\$, there are \$\displaystyle 5000x\$ fewer ticket sales.
. . They will sell: .\$\displaystyle (45000 - 5000x)\$ tickets.

So \$\displaystyle (45000-5000x)\$ tickets are sold at \$\displaystyle (10+x)\$ dollars each.

The revenue from ticket sales is: .\$\displaystyle R_1 \;=\;(10+x)(45000-5000x)\$ dollars.

The same \$\displaystyle (45000-5000x)\$ ticketholders will spend \$5 each on concessions.

The revenue from concessions is: .\$\displaystyle R_2 \;=\;5(45000-5000x)\$ dollars.

Hence, the Total Revenue is: .\$\displaystyle R \;=\;(10+x)(45000-5000x) + 5(45000-5000x)\$ dollars.

. . and that is the function we must maximize.