1. The diameter of a pot hole is 6 cm. Over the course of 3 weeks the diameter increases to 15 cm. Write an explicit algebraic expression that represents this exponential relationship where x is the number of weeks. Round to two decimal places

Question

A. F(x)=15(0.4)^x
B. F(x)=6(1.36)^x
C. F(x)=15(0.74)^x
D. F(x)=6(2.5)^x
2. Sully had to solve the following homework problem. He asks his friend Mike to check his work after he finishes. Problem: At the end of January, a company's profits were $422,000. At the end of May, the company's profits were down to $368,000. Write an explicit algebraic expression that represents this exponential relationship where x is the number of months since January. Round to two decimal places. Sully's Answer: f(x)=422,000(1.03)^x Mike looks at Sully's answer first without looking at the calculations and says the answer must be wrong. Is Mike correct? Explain.
A. Mike is not correct. Sully’s answer is the correct answer.
B. Mike is not correct. He would have to check Sully's work to determine if he made any errors.
C. Mike is correct. It is given that the company's profits decreases, so the rate would have to be less than 1 to indicate exponential decay.
D. Mike is correct. Sully substituted the incorrect constant and should have used $368,000 in his answer.
3. an experimental physicist created a sample of technetium-99 and recorded the amount of the isotope at equally spaced time intervals to produce the following table.
N 0 1 2 3 4
F(n) 10 8.911 7.940 7.075 6.295
Write a recursive model for the amount of isotope remaining at the nth time interval.
A. f(n)=0.891 f(n−1), f(0)=10
B. f(n)=1.089 f(n−1), f(0)=10
C. f(n)=−1.089 f(n−1), f(0)=10
D. f(n)=1.122 f(n−1), f(0)=10
4. A bank offers a savings account with a yearly interest rate of 6% compounded monthly. Create a recursive model for the yearly balance for an account that starts with $500. Let n=0 represent the moment the bank account opened. Note: 6% compounded monthly means you need to figure out what the annual interest rate is first!
A. f(n)=1.005 f(n−1), f(0)=500
B. f(n)=1.06 f(n−1), f(0)=500
C. f(n)=1.061678 f(n−1), f(0)=500
D. f(n)=2.012196 f(n−1), f(0)=500
5. A bacterial population has a doubling time of 1.8 hours. How can you calculate the base for an exponential model f(x)=ab^x if x is expressed in hours?
A. The base is b=2^1.8
B. The base is b=1.8 .
C. The base is b= 2/1.8
D. The base is b=2^(1/1.8)

Answer 1

1. The correct answer is A. F(x)=15(0.4)^x. Since the diameter increase is exponential, the base of the exponential expression must be between 0 and 1, which is 0.4 in this case. The starting diameter is 6 cm, so it is multiplied by the exponential factor (0.4) raised to the power of the number of weeks (x).

2. The correct answer is B. Mike is not correct. Sully's answer is the correct answer. Since the company's profits are decreasing, the exponential base must be less than 1 to indicate exponential decay. In Sully's answer, the base is 1.03, which is less than 1.

3. The correct answer is A. f(n)=0.891 f(n−1), f(0)=10. The table shows that the amount of the isotope remaining at each time interval is multiplied by a factor of 0.891 compared to the previous interval. This recursive relationship is represented by f(n)=0.891 f(n−1), where f(0)=10 represents the initial amount.

4. The correct answer is A. f(n)=1.005 f(n−1), f(0)=500. The yearly interest rate of 6% compounded monthly means that the interest is applied 12 times a year, and the monthly interest rate is (1 + 0.06/12). Therefore, the recursive model for the yearly balance is f(n)=1.005 f(n−1), where f(0)=500 represents the initial balance.

5. The correct answer is D. The base is b=2^(1/1.8). The doubling time of 1.8 hours indicates that the population is multiplying by a factor of 2 every 1.8 hours. To find the base for the exponential model, we take the 1/1.8 power of 2, which is the same as taking the 1.8th root of 2. Therefore, the base is b=2^(1/1.8).