word problem HELPPPP

Question

the annual yield per walnut tree is fairly constant at 60 pounds per tree when the number of trees per acer is 20 or less. for each additional tree over 20, the annual yield per tree for all trees on the acre decreases by 2 pounds due to overcrowding. how many walnut trees should be planted to maximize the annual yield for the acre? what is the maximum number of pounds of walnuts per acre?  

 

thanks for the help, i need it really quick 

Answer 1

This is a very typical parabola problem.

So, if you have 20 trees per acre, each tree produces 60 pounds, so you have a total of 60*20=1200 lbs.

 

For every tree increase, you lose 2 pounds per tree....so if you add x trees, you lose 2x pounds....so you can write the following function:

 

P(x) = (20 + x)*(60 - 2x) for the pounds of nuts based on trees added to 20 and per tree output going down by 2 lbs.

 

So P(x) = 1200 + 20x - 2x² 

 

If you know about derivatives, you can do it that way, but since this is a parabola, the Maximum (or minimum) must be on the axis of symmetry. The formula for the axis of symmetry is x=-b/2a, where a and b come from y=ax²+bx+c.

 

So our a = -2 and our b=20, so our axis of symmetry is x=-20/-4 = 5. You can now plug this into your function to find out the maximum output comes from planting 5 extra trees for a total of 1200+20(5)-2(5)²=1250 pounds.

 

That's the interpretation of the turning point being (5,1250).

 

Hope this helps.

Answer 2

Read the problem carefully. The quantity to be maximized is walnut yield per acre.

Now use the conditions of the problem to express annual yield of pounds of walnuts per acre as a function in one variable. The total annual yield of pounds of walnuts per acre is the product of the number of walnut trees per acre and the pounds of walnuts per tree. Let x be the number of walnut trees and let y be the pounds of walnuts per tree.

An equation for walnut yield per​ acre, W, is W = xy.

Determine an expression for y in terms of x. The value of y is modeled by a linear function. To write the equation of this​ line, first find the slope m. Note that an increase of one walnut tree corresponds to a decrease of 2 pounds of walnuts.

m = −2/1 = −2

The condition that 60 pounds of walnuts is produced per tree when there are 20 walnut trees in the acre corresponds to the point (20​, 60​). Use this point and the slope m = −2 in the​ point-slope form of a line to write the equation for the pounds of walnuts per tree.

y − y₁ = m (x − x₁) This is the​ point-slope form of a line.

y − 60 = −2 (x − 20) Substitute the known values.

y − 60 = −2x + 40 Use the distributive property.

Isolate y on the left side of the equation to write the equation in​ slope-intercept form.

y − 60 = −2x + 40

y = −2x + 100 Add 160 to both sides of the equation and simplify.

Substitute the value of y into the equation for annual yield of pounds of walnuts per acre W=xy so that the equation is now a function of x only.

W = xy

W(x) = x(−2x + 100) Substitute −2x + 100 for y.

Now write the function​ W(x) in the form f(x) = ax² + bx + c.

W(x) = x(−2x + 100)

W(x) = −2x² + 100x Use the distributive property.

Notice that for this​ function, c=0. Identify the values of a and b.

a = −2 and b = 100

Calculate −b/2a. Since a < 0​, W has a maximum at x = −b/2a.

x = −100/2(−2) Substitute the values for a and b.

x = 25 Simplify.

​

Therefore, plant 25 walnut trees per acre to maximize annual yield for the acre.

This maximum annual yield is given by W(−b/2a)​, or, in this​ case, W(25). Find this value.

W(x) = −2x² + 100x

W(25) = −2(25)² + 100(25) Substitute 25 for x.

W(25) = 1250 Simplify.

Since the number of walnut trees that maximizes annual yield for the acre is 25 trees, the maximum annual yield is 1250 pounds of walnuts per acre.