Question

If all goes according to plan, the function A(t)=200(1.1)^tA(t)=200(1.1) t A, left parenthesis, t, right parenthesis, equals, 200, left parenthesis, 1, point, 1, right parenthesis, start superscript, t, end superscript will model the amount of potatoes, AAA, in bushels, produced by Burian's farm ttt years from now, and the function R(t)=5000(1.111)^tR(t)=5000(1.111) t R, left parenthesis, t, right parenthesis, equals, 5000, left parenthesis, 1, point, 111, right parenthesis, start superscript, t, end superscript will model the revenue, RRR, in dollars, earned from selling these potatoes. Let PPP be the proposed price of a single bushel of potatoes ttt years from now. Write a formula for P(t)P(t)P, left parenthesis, t, right parenthesis in terms of A(t)A(t)A, left parenthesis, t, right parenthesis and R(t)R(t)R, left parenthesis, t, right parenthesis.

Answer 1

Answer:

• The formula for P(t) in terms of R(t) and A(t) is:

        $P(t)=\frac{R(t)}{A(t)} =\frac{5000(1.111)^t}{200(1.1)^t}$

• It can be simplified in terms of t as:

         $P(t)=25(1.01)^t$

           

Explanation:

I will rewrite the question removing the errors, for better understanding:

• If all goes according to plan, the function A(t)=200(1.1)^t will model the amount of potatoes, A, in bushels, produced by Burian's farm t years from now, and the function R(t)=5000(1.111)^t will model the revenue, R, in dollars, earned from selling these potatoes. Let P be the proposed price of a single bushel of potatoes t years from now. Write a formula for P(t).

You must depart from the equation that relates revenue (R), price (P), and number of items sold.

• Revenue = Price × Number of items

• R (t) = P(t) × A(t)

There you see that you can solve for P(t), which is the unknown function:

• P(t) = R(t) / A(t)

Substituting you get:

• $P(t)=\frac{R(t)}{A(t)} =\frac{5000(1.111)^t}{200(1.1)^t}$

Divide the coefficients (5,000 / 200) and apply the power properties for the quotients of powers with the same power:

• $P(t)=25(1.111/1.1)^t=25(1.01)^t$