The Correct Answer and Explanation is:
Correct Answer: 7.5%
Explanation
The provided equation, y = 35(1.075)^x, is an example of an exponential growth function. To understand how to find the rate of growth, we can compare it to the standard formula for exponential growth, which is y = a(1 + r)^x.
In this general formula:
- y represents the final amount after a certain number of time periods.
- a is the initial or starting amount.
- r is the rate of growth, expressed as a decimal.
- x represents the number of time periods that have passed.
- The term (1 + r) is known as the growth factor. It represents the original amount (the 1) plus the new growth (the r) for each time period.
Now, let’s match the components of the given equation, y = 35(1.075)^x, to the standard formula:
- The initial amount, a, is 35.
- The growth factor, (1 + r), is 1.075.
The question asks for the “rate of growth,” which corresponds to the variable r. We can find r by using the growth factor from our equation. We set up the following simple equation:
1 + r = 1.075
To solve for r, we subtract 1 from both sides of the equation:
r = 1.075 – 1
r = 0.075
This gives us the rate of growth as a decimal. To express this rate as a percentage, which is the more common format, we multiply the decimal by 100:
0.075 * 100 = 7.5%
Therefore, the rate of growth for the function y = 35(1.075)^x is 7.5%. This means that for each unit increase in x, the value of y increases by 7.5% of its previous value.
