Question

Two princesses, Nat and Kat, are gifted with a savings account from their royal parents, Matt and Pat, when they turn 5 years old. The royal parents put a specific amount into each account annually such that Nat’s account follows a linear model. Nat’s savings account has an initial balance of when it is opened and each year on her birthday, Nat’s parents put a constant amount into the account and never remove any money. Her account has $6200 when she is 7 and when Nat is 12, she has $21200 in her account.
a. Write a linear function N(t) for the amount of money, N, that Nat has in her account ‘t’ years after her 5th birthday.
b. Kat’s account doesn’t follow a linear model, but rather a quadratic model. On her 5th birthday, her parents open the account and then add money annually just like with Nat. Kat’s parents plan is to add money so that her account maxes out at $44100 on her 25th birthday. At the age of 14, she has $32000. Write a quadratic function K(t) for the amount of money, K, in Kat’s account ‘t’ years after her 5th birthday.
c. What age will both Nat and Kat have the same amount in their account?

Answer 1

Answers and Step-by-step explanations:

(a) We want to find the slope-intercept equation (y = mx + b), where m is the slope and b is the y-intercept. We already have two points: (2, 6200) and (7, 21200). The slope is change in y over change in x so:

m = (21200 - 6200) / (7 - 2) = 3000

Now, we have y = 3000x + b. Plug in 6200 for y and 2 for x to solve for b:

y = 3000x + b

6200 = 3000 * 2 + b

b = 200

Nat's equation is y = 3000t + 200.

(b) We want a quadratic formula in vertex form because the problem says that Kat's account maxes out at 44100 on her 25th birthday, indicating that's the vertex. Our vertex point is (20, 44100); note that it's 20 and not 25 because she starts the account at age 5, not age 0. The generic vertex form of a quadratic is:

y = a(x - h)² + k, where (h, k) is the vertex

Plug in our vertex:

y = a(x - 20)² + 44100

We also have the point (9, 32000), so put that in to find a:

32000 = a(9 - 20)² + 44100

-12100 = a * 121

a = -100

So Kat's equation is y = -100(t - 20)² + 44100.

(c) In order to find the age when they have the same amount, we set the two equations equal:

3000t + 200 = -100(t - 20)² + 44100

-100(t² - 40t + 400) + 44100 = 3000t + 200

-100t² + 4000t - 40000 + 44100 - 3000t - 200 = 0

-100t² + 1000t + 3900 = 0

t² - 10t - 39 = 0

(t - 13)(t + 3) = 0

t = 13 or t = -3   --   obviously, years can't be negative, so t = 13.

Note, however, that t is the number of years after Kat and Nat are 5, so we need to add 13 to 5:

13 + 5 = 18

The age is 18.

Answer 2

A: $N(t) = 3000t +200$

B: $y = -100(x-20)^2+44100$

C: When they turn 18.

A. Write a linear function N(t) for the amount of money, N, that Nat has in her account ‘t’ years after her 5th birthday:

You can determine a linear equation using the two coordinates provided:

(2, 6200) and (5, 21200).

To do this, first find the slope of the graph. You can do this by using the expression

$\frac{y2-y1}{x2-x1}$, where the coordinate (2, 6200) is 1 and (7, 21200) is 2:

$\frac{21200-6200}{12-7}\\\\\frac{15000}{5}$

The slope is equal to 3000. Now, you need to find the y-intercept. Plug in 2 for x and see what the y intercept would need to be to equal 6200:

6200 = 3000 * 2 + b

6200 = 6000 + b

b = 200

Thus, your equation for Nat's bank account is $N(t) = 3000t + 200$

B: Kat’s account doesn’t follow a linear model, but rather a quadratic model. On her 5th birthday, her parents open the account and then add money annually just like with Nat. Kat’s parents plan is to add money so that her account maxes out at $44100 on her 25th birthday. At the age of 14, she has $32000. Write a quadratic function K(t) for the amount of money, K, in Kat’s account ‘t’ years after her 5th birthday:

The question supplies you with the vertex to the parabola, (20, 44100). This is the point at which the maximum value of the function is reached.

It also supplies you with a point on the parabola, (9, 32000).

You can use the vertex form of the quadratic equation to find the formula for Kat's bank account:

$y=a(x-h)^2+k$, where (h, k) is the vertex.

Substituting the vertex in for that equation, we get:

$y=a(x-20)^2+44100$

Now, you can substitute the point (14, 32000) in for x and y to solve for a:

$32000=a(9-20)^2+44100\\\\-12100 = 121a\\a = -100$

Thus, the equation is $y = -100(x-20)^2+44100$.

C. What age will both Nat and Kat have the same amount in their account? 18 years old.

To find what age each account will be equal, set the equations equal to each other:

$-100(x-20)^2+44100 = 3000x +200$

Then, you can solve for x:

$-100(x-13)(x+3)=0$

Solving for x, we notice that the accounts equal each other when Nat and Kat are -3 or 13. However, Nat's function is only specified to begin after she turns 5, so add 5 to 13 to get 18.

They will both have the same money in their accounts when they turn 18.