Application of Quadradic Function

[snix]

Please help! I've spent hours trying to find the solution to this problem. I have to take a quiz using this concept tomorrow and I'm lost!

The manager of a peach orchard is trying to decide when to arrange for picking the peaches. If they are picked now, the average yield per tree will be 100lbs, which can be sold for 40 cents per pound. Past experience shows that the yield per tree will increase about 5 pounds per week, while the price will decrease about 2 cents per pound per week.

#a) let x represent the number of weeks that the manager should wait. Find the price per pound.

my answer was: .40-.02x (the book says the answer is 40-2x. However, in a simular problem, the book has the answer in decimals.)
Which is right?

#b) find the number of lbs. per tree.

100+5x (the book agrees with this answer)

#c) Find the total revenue from a tree.

R(x) = (100+5x)(40-2x) - Using the books answer to #a)
= 4000-200x+200x-10x²
= 4000-10x²

The book agrees with this answer

#d) When should the peaches be picked in order to produce themaximum revenue?
#e) What is the maximum revenue?

I think I need to use the vertex to answer these questions; however, I'm having difficulty with this step. Here's what I have so far:

R(x)=(100+5x)(40-2x) Using the books answer to a)
= 4000-200x+200x-10x²
= 4000-10x²
= -10x²+4000
= -10(x²-400) - I don't know how to complete the square from this point. The book shows the answer should be (0,40) but I don't know how to get there.

 

[soroban]

Hello, snix!

The manager of a peach orchard is trying to decide when to arrange for picking the peaches.
If they are picked now, the average yield per tree will be 100 lbs, which can be sold for 40 cents per pound.
Past experience shows that the yield per tree will increase about 5 pounds per week,
. . while the price will decrease about 2 cents per pound per week.

a) Let \(\displaystyle x\) represent the number of weeks that the manager should wait.
. . Find the price per pound.

My answer was: .0.40 -0.02x . Your answer is in dollars.

The book says the answer is 40 - 2x.
However, in a simular problem, the book has the answer in decimals.
Which is right? .The book



b) Find the number of lbs per tree.

100 + 5x . (The book agrees with this answer)



c) Find the total revenue from a tree.

R(x) = (100 + 5x)(40 - 2x) . . . Using the books answer to a)
. . . = 4000 - 200x + 200x - 10x²
. . . = 4000 - 10x²

The book agrees with this answer

d) When should the peaches be picked in order to produce the maximum revenue?

Given the parabola: .\(\displaystyle y \:=\:ax^2 + bx + c\), the vertex is at: .\(\displaystyle x \:=\:\dfrac{\text{-}b}{2a}\)

Our parabola is: .\(\displaystyle y \:=\:4000 - 10x^2 \:=\-10)x^2 + (0)x + 4000\)

Hence, the vertex is at: .\(\displaystyle x \:=\:\dfrac{-0}{2(-10)} \:=\:0\)

The peaches should be picked now.

e) What is the maximum revenue?

\(\displaystyle R(0) \:=\:4000 - 10(0^2) \:=\:4000\text{ cents per tree}\)

 

[Deleted member 4993]

Please help! I've spent hours trying to find the solution to this problem. I have to take a quiz using this concept tomorrow and I'm lost!

The manager of a peach orchard is trying to decide when to arrange for picking the peaches. If they are picked now, the average yield per tree will be 100lbs, which can be sold for 40 cents per pound. Past experience shows that the yield per tree will increase about 5 pounds per week, while the price will decrease about 2 cents per pound per week.

#a) let x represent the number of weeks that the manager should wait. Find the price per pound.

my answer was: .40-.02x (the book says the answer is 40-2x. However, in a simular problem, the book has the answer in decimals.)
Which is right?

#b) find the number of lbs. per tree.

100+5x (the book agrees with this answer)

#c) Find the total revenue from a tree.

R(x) = (100+5x)(40-2x) - Using the books answer to #a)
= 4000-200x+200x-10x²
= 4000-10x²

The book agrees with this answer

#d) When should the peaches be picked in order to produce themaximum revenue?
#e) What is the maximum revenue?

I think I need to use the vertex to answer these questions; however, I'm having difficulty with this step. Here's what I have so far:

R(x)=(100+5x)(40-2x) Using the books answer to a)
= 4000-200x+200x-10x²
= 4000-10x²
= -10x²+4000
= -10(x²-400) - I don't know how to complete the square from this point. The book shows the answer should be (0,40) but I don't know how to get there.

Assuming you have done the arithmatic correctly,

-10(x² - 400) = -10(x² - 20²) = -10(x - 20)(x + 20)

 

[snix]

Thanks for the Help!

I missed that formula for the Vertex. Thanks!