min. length of fencing; best ticket price; average cost

[whytehaze]

1. A farmer wants to create a 300 square foot rectangular grazing pen for his sheep. The barn will form one side of the pen, but he will will have to buy fencing for the other three sides. Find the minimum length of fencing that will be required.

2. A baseball stadium holds 20,000 people. At a selling price of $15 per ticket, the stadium fills to capacity. For each $1 in ticket price, 500 fewer tickers are sold.

a.Find the ticket price that maximizes revenue from ticket sales.
b. Find the maximum possible revenue from ticket sales in the stadium.

3. The cost to produce q goods is given by C(q)=2000+50q+2q^(2) for 0<q<100.
a. Find the average cost if 70 items are sold.
b. Find the minimum average cost.
c. Find the maximum profit if each item is sold for $350.

 

[galactus]

Re: Calc Help

For #1, you don't even need calc. Actually, you don't for any. But use it if required to.

These problems are very cliche.

#1. The perimeter is P=2W+l........[1]. Because we only have 3 sides.

The area is A=lw=300.........[2].

Solve [2] for l or w and sub into [1]. You will then have a quadratic you can solve by differentiating or using -b/(2a).

 

[stapel]

What are your thoughts? What have you tried? How far did you get? Where are you stuck?

1. A farmer wants to create a 300 square foot rectangular grazing pen for his sheep. The barn will form one side of the pen, but he will will have to buy fencing for the other three sides. Find the minimum length of fencing that will be required.

Back in algebra, you did plenty of "maximizing the area of a field" stuff in quadratics, where you found the vertex of the parabola, etc, etc. Start this one in the same manner, by drawing a picture, defining two variables, and writing down the "area" and "perimeter" formulas. Solve the one equation for which you have a value for one of the variables, and plug this into the other formula to get your function. Then differentiate, etc.

2. A baseball stadium holds 20,000 people. At a selling price of $15 per ticket, the stadium fills to capacity. For each $1 in ticket price, 500 fewer tickers are sold.
a.Find the ticket price that maximizes revenue from ticket sales.
b. Find the maximum possible revenue from ticket sales in the stadium.

Start making a table, just like back in algebra. You are making changes in the price by one dollar, with each change leading to one five-hundred drop in sales. So what should your variable stand for?

When you have zero changes, what is the price? What is the volume? What is the total income?
When you have one change, what is the price? What is the volume? What is the total income?
When you have two changes, what is the price? What is the volume? What is the total income?

Keep going until you find the pattern. Then create the quadratic model, and find the vertex. Interpret this as you learned back in algebra.

13. The cost to produce q goods is given by C(q)=2000+50q+2q^(2) for 0<q<100.
a. Find the average cost if 70 items are sold.
b. Find the minimum average cost.
c. Find the maximum profit if each item is sold for $350.

a) Use algebra and pre-algebra: What is the number of items? What is the total cost for this number of items? Then what is the average cost?
b) What operation did you use in (a) to find the average cost? What then should be your average-cost function? Differentiate to find the minimizing value.
c) How does profit relate to total costs and total income? Use this to create your profit function. Differentiate to find the maximizing value.

If you get stuck, please reply with a clear listing of your work and reasoning so far. Thank you!

Eliz.