Question

2.The amount of fertilizer in a landscaping company's warehouse decreases at a rate of 3% per week. The amount of fertilizer in the warehouse was originally 78,000 cubic yards. What function models the amount of fertilizer in cubic yards left after w weeks? Exponential Growth: y = a*b* where a initial amount and b is the growth factor. Sarah identified the initial amount as 78000 and the rate as 3%. She wrote her equation as: y = 78,000 (3)*.

Answer 1

ANSWER

$y=78000\cdot0.97^w$

EXPLANATION

We want to find the function that models the amount of fertilizer after w weeks.

The general form of an exponential function is written as:

$y=a\cdot b^x^{}$

but it can be re-written as:

$y=a(1\pm r)^x^{}$

+ is used when it an exponential growth

- is used when it is an exponential decay

Note: r stands for rate of growth/decay

From the question, we see that there is an exponential decay (decrease).

This means that the function, we need to use is:

$y=a(1-r)^x$

The mistake that Sarah made was that she used her rate directly as 3% (i.e. she made b = 3) into the general function which is wrong.

She should have instead used the alternate function given above to find the function.

Therefore, the correct function is:

$\begin{gathered} y=78000(1-\frac{3}{100})^w \\ y=78000(1-0.03)^w \\ y=78000\cdot0.97^w \end{gathered}$

where w represents the number of weeks