Quadratic inequality....?

Question

the problems are word problems, and if you have trouble with word problems, you will know my delema, when it said that they had to be converted into Quadratic inequality, and solved... i knew what both of these where, and i was just like "............what??? you want me to do what?????"
so in hopes of one of you knowing how to do this, or knowing someone that does, i am sending along the example questions....

1) if 15 persons sign up for a boat trip, the price per person will be $50. The ticket price is reduced by $2 for each additional passanger beyond 15. How many passengers will produce the maximum income, and what is the maximum income??


2) 400 people will attend a concert if tickets cost $10 each. Attendance will decreace by 20 people for each $1 increased in the price. What ticket price will yeild the maximum income, and what is the maximum income from ticket sales?


3) and apple orchard has 20 trees per acre and each tree produces an average of 200 apples. For each additional tree planted per acre, the average yeil will be reduced by 5 apples per tree. how many trees per acre will produce the maximum number of apples, and whtat is the maximum number of apples per acre?

Answer 1

1. Step 1. Write an equation.
You make $x(50-2(x-15)) where x is the # of people.

Why?
the first x is the # of passengers.
The rest is determining the cost: $50 initial - 2 * # of passengers over 15 = $50 - 2*(x-15)

Step 2: Distribute parenthesis:
income = x(50 - 2x + 30)
income = 80x - 2x^2

Step 3: To find the maximum income, take the derivative of the income, and solve for 0 (A parabola's vertex is at the point where slope = 0).
f(x) = 80x - 2x^2
f'(x) = 80 - 4x
4x = 80
x = 20 (solution)
Income = x(50-2(x-15))
= 20(50-2(20-15)
= 20(50-10) = 20 * 40 = $800 (solution)

Check:
at $20, the ticket price is 50-2*5 = 40, thus the income is 20 * 40 = $800
at $19, the ticket price is 50-2*4 = 42, thus the income is 19 * 42 = $798
at $21, the ticket price is 50-2*6 = 40, thus the income is 21 * 38 = $798

2.) You make x(400-20(x-10)) where x is the cost.

Distribute:
x(400 - 20x + 200)
f(x) = 600x - 20x^2

Take derivative:
f'(x) = 600 - 40x
40x = 600
x = $15 (solution)
Income: x(400-20(x-10))
= 15(400-20(15-10))
= 15(400-20(5))
= 15(300) = $4500
Check:
at $15, the attendance is 400 - 20 * 5 = 300, thus the income is 15 * 300 = $4500
at $16, the attendance is 400 - 20 * 6 = 280, thus the income is 16 * 280 = $4480
at $14, the attendance is 400 - 20 * 4 = 320, thus the income is 14 * 320 = $4480

3.) You produce x(200-5(x-20)) apples.

Distribute:
x(200 - 5x + 100)
f(x) = 300x - 5x^2

Derivative:
f'(x) = 300 - 10x
10x = 300
x = 30 trees (solution)
Total apples = x(200-5(x-20))
= 30(200-5(30-20))
= 30(200 - 50)
= 30 * 150 = 4500 apples (solution)

Check:
at 30 trees/acre, the yield is 200 - 5 * 10 = 150 apples/tree, thus total yield is 30 * 150 = 4500
at 29 trees/acre, the yield is 200 - 5 * 9 = 155 apples/tree, thus total yield is 29 * 155 = 4495
at 31 trees/acre, the yield is 200 - 5 * 11 = 145 apples/tree, thus total yield is 31 * 145 = 4495

Answer 2

2) 300 people will give max income of $4500.
1) It depends on the capacity of the boat.