At age 15, Nan is twice as tall as her 5-year-old brother Dan, but on Dan's 21st birthday, they find that he is 6 inches taller. Explain why there must have been a time when they were exactly the same height
So you are studying the mean value theorem....
If growth is continual, even at different rates, if Dan starts out shorter than her, and ends up taller, at some time he must have been the same height...how else can he pass her?
I believe we have to express it as a function using f(x) and g(x)....not sure.
let N(t) be nan's height
and D(t) be dan's height.
N(a)>D(a) and
D(b)>N(b) given
Let N'be Nan's growth rate and D' be Dans growth rate.
N(t)=N(a)+N'*(t-a)so
N(b)=N(a)+N'*(b-a). Simillary,
D(b)=D(a)+D'*(b-a)
N(a)-N(b)>D(a)-D(b)
N(a)-N(a)-N'(b-a)>D(a)-D(a)-D'(b-a
or N'<D' which says that Dan's growth rate is greater than Nans.
Is there a time which N(t)=D(t)?
iF so, then
D(a)+D'*(t-a)=N(a)+N'(t-a)
(t-a)= (N(a)-D(a))/((D'-N')
and t= ( ) + a
Since this time computes for all N', D', such that D'>N', this time must exist.
take time 0 when Dan is 5, then t = 16 when Dan is 21
D(t) = Dan height (always positive)
N(t) = Nan height (always positive)
N(0) = 2D(0)
N(16) = D(16)-6
These two curves cross between t = 0 and t = 16. The heights are the same at the intersection.
Of course there may have been more than one time when they were equal height.
To determine why there must have been a time when Nan and Dan were the same height, let's break down the information provided:
1. At age 15, Nan is twice as tall as her 5-year-old brother Dan.
2. On Dan's 21st birthday, they find that he is 6 inches taller.
First, let's establish the key ages to analyze the situation:
Let N represent Nan's age, and D represent Dan's age.
- At age 15, Nan is twice as tall as Dan (who is 5 years old): N = 15, D = 5
- On Dan's 21st birthday, they find that he is 6 inches taller: N = ?, D = 21
Using this information, let's calculate their heights at different ages:
Height at age 15:
Nan's height = 2 * Dan's height
N15 = 2 * D5
Height on Dan's 21st birthday:
Nan's height = Dan's height + 6 inches
N? = D21 + 6
Now, let's analyze the problem systematically by converting their ages into the number of years since birth. For instance, Nan was born 15 years ago, so her age is N - 15. Similarly, Dan's age is D - 5.
Therefore, we can rewrite the equations as follows:
N - 15 = 2 * (D - 5) (Equation 1)
N - ? = D - 21 + 6 (Equation 2)
We now have two equations that describe the relationship between Nan's and Dan's ages and heights at different times.
To find when Nan and Dan were the same height, we need to equate their heights (N = D) and solve the equations for N and D:
N - 15 = 2 * (D - 5)
N - ? = D - 21 + 6
Let's substitute the value of N from Equation 1 into Equation 2:
2 * (D - 5) - 15 = D - 21 + 6
2D - 10 - 15 = D - 15
D - 5 = D - 15
Simplifying the equation further, we get:
-5 = -15
This equation is not possible since -5 is not equal to -15. Therefore, there is no solution to the equations given. Based on the information provided, it appears that there was no time when Nan and Dan were exactly the same height.
It is important to note that this conclusion is based on the information provided. If there are any additional details or if any information is missing, the conclusion may change.