Question

Rapunzel's hair is currently 110 inches long. She needs her hair to be at least 150 inches long in order to support a male suitor climbing it. She is currently using a magical conditioner that results in growth of 5% per week. A traveling witch offers to cast a spell on her hair that will result in growth of 7.5% per week, but will require her to trim 10 inches off in order for the spell to be effective. 1. Write an exponential function to represent the length of Rapunzel's hair after x number of weeks if she continues to use the magical conditioner and doesn't take the witch's offer. 2. Write an exponential function to represent the length of Rapunzel's hair after number of weeks if she lets the witch trim her hair and cast the spell. 3. After four weeks, what will the length of her hair be with each option? Will her hair be long enough to support a male suitor? 4. After six weeks, what will the length of her hair be with each option Should she let the witch cast her spell?

Answer 1

Answer: 1. H₁(x) = 110.
`(1.05)^(x)`

2. H₂(x) = 100.
`(1.075)^(x)`

3. Option 1: H₁(4) = 133.7 in

Option 2: H₂(4) = 133.5 in

4. H₁(6) = 147.4 in

H₂(6) = 154.3 in

Step-by-step explanation: An exponential function is defined as: A = b(f)ˣ, where A is the final value; b is the initial value; f is a constant factor; x is dependent variable of the function.

1. With magical conditioner, the initial length of the hair is 110 inches and the growth factor is 5%:

(1 + 5/100) = (1.05)

The function will be:

H₁(x) = 110.
`(1.05)^(x)`

2. If she accepts the witch's proposal, the initial length will be 100, due to the trimming, and growth factor is:

(1 + 7.5/100) = (1.075)

The function is:

H₂(x) = 100
`(1.075)^(x)`

3. After 4 weeks:

Option 1:

H₁ (4) = 110.
`(1.05)^(4)`

H₁ (4) = 133.7 in

Option 2:

H₂(4) = 100.
`(1.075)^(4)`

H₂(4) = 133.5 in

No, in either option, Rapunzel's hair won't be long enough to support a male suitor.

4. After 6 weeks:

Option 1:

H₁ (6) = 110.
`(1.05)^(6)`

H₁ (6) = 147.4 in

Option 2:

H₂(6) = 100.
`(1.075)^(6)`

H₂(6) = 154.3 in

Yes, she should let the witch cast her spell because, if she does it, Rapunzel's hair will be long enough to support a male suitor after 6 weeks.