Let f(t) = Q0at = Q0(1 + r)t.

f(7) = 75.94 and f(9) = 91.89
(a) Find the base, a. (Round your answer to two decimal place.)
a =


(b) Find the percentage growth rate, r. (Round your answer to the nearest percent.)
r = %

I honestly have no idea how to approach these problems. I wrote out the different equations:
75.94 = Q0a^7 and
91.89 = Q0a^9 but I don't know where to go from here for the first part.

For the second part of the question, I'm assuming that you plug in the values to the r equation so it would be:

75.94=Q0(1+r)^7 and
91.89=Q0(1+r)^9 but once again, I do not know where to go from here.

Should I use a system of equations?

I will try to reverse engineer what you are doing because I am not sure I understand you

Say we are doing a compound interest problem. Yearly interest rate as a decimal is r
For example 5% is .05
then every year I multiply what I have is the amount from the year before + r times the amount from the year before
or
Yn+1 = Yn + r Yn = Yn (1+r)

if Y0 = a
then Y1 = a(1+r)
and Y2 = a(1+r)(1+r) = a(1+r)^2
Yn = a (1+r)^n
so
Y9 = 91.89 = Y7(1+r)(1+r)
or 91.89 = 75.94 (1+r)^2
so
(1+r)^2 = 1.21
1+r = 1.10
r = .1 or 10 percent
now for a
y9 = 91.89 = a(1+r)^9 = a (1.1)^9
so a = 91.89/2.358 = 38.97
so I started with $38.97
compounded at 10% yearly
and after nine years had 91.89

To solve the problem, you can indeed use a system of equations. Let's proceed step by step.

(a) Finding the base, a:

We have the equation: 75.94 = Q0a^7

To solve for a, we need to eliminate Q0. Since it is a constant, we can divide the two equations:

(91.89 / 75.94) = (Q0a^9 / Q0a^7)

Simplifying, we get:

(91.89 / 75.94) = a^2

Now, we can solve for a by taking the square root of both sides:

a = sqrt(91.89 / 75.94)

Rounding to two decimal places, we find:

a ≈ 1.07

So, the base, a, is approximately 1.07.

(b) Finding the percentage growth rate, r:

We have the equation: 75.94 = Q0(1 + r)^7

Similarly, we can divide the two equations:

(91.89 / 75.94) = (Q0(1 + r)^9 / Q0(1 + r)^7)

Simplifying, we get:

(91.89 / 75.94) = (1 + r)^2

Now, we need to solve for (1 + r). Taking the square root of both sides, we get:

sqrt(91.89 / 75.94) ≈ (1 + r)

Subtracting 1 from both sides, we have:

sqrt(91.89 / 75.94) - 1 ≈ r

To find the percentage growth rate, we multiply r by 100:

r ≈ (sqrt(91.89 / 75.94) - 1) * 100

Rounding to the nearest percent, we find:

r ≈ 4%

So, the percentage growth rate, r, is approximately 4%.

In summary:

(a) The base, a, is approximately 1.07.
(b) The percentage growth rate, r, is approximately 4%.

For the first part of the question, you have correctly written out the equations:

75.94 = Q0 * a^7
91.89 = Q0 * a^9

To solve for the base, "a", you can divide the second equation by the first equation:

91.89 / 75.94 = (Q0 * a^9) / (Q0 * a^7)

Simplifying, you get:

1.209 = a^2

Now, take the square root of both sides:

√(1.209) = √(a^2)

a ≈ 1.10 (rounded to two decimal places)

So, the base "a" is approximately 1.10.

For the second part of the question, you need to find the percentage growth rate, "r".

Using the information from the original equation: f(t) = Q0 * a^t

Plug in the given values:

75.94 = Q0 * a^7
91.89 = Q0 * a^9

Divide the second equation by the first equation:

91.89 / 75.94 = (Q0 * a^9) / (Q0 * a^7)

Simplifying, you get:

1.209 ≈ a^2

Now, take the square root of both sides:

√1.209 ≈ √(a^2)

1.100 ≈ a

Now, substitute this value of "a" back into the equation:

75.94 = Q0 * (1 + r)^7

Rearrange the equation to isolate (1 + r)^7:

(1 + r)^7 ≈ 75.94 / Q0

Similarly, substitute the value of "a" into the second equation:

91.89 = Q0 * (1 + r)^9

Rearrange to isolate (1 + r)^9:

(1 + r)^9 ≈ 91.89 / Q0

Now, you have two equations with (1 + r)^7 and (1 + r)^9. Take the ninth root of the first equation and the seventh root of the second equation:

√(1 + r)^7 ≈ √(75.94 / Q0)

∛(1 + r)^9 ≈ ∛(91.89 / Q0)

Simplifying:

1 + r ≈ √(75.94 / Q0)

1 + r ≈ ∛(91.89 / Q0)

Subtract 1 from both sides:

r ≈ √(75.94 / Q0) - 1

r ≈ ∛(91.89 / Q0) - 1

Now, plug in the value of "a" (1.10) into the expressions:

r ≈ √(75.94 / 1.10) - 1 ≈ 0.028

r ≈ ∛(91.89 / 1.10) - 1 ≈ 0.055

Lastly, convert the percentage growth rate to the nearest percent:

r ≈ 0.028 ≈ 2.8%

r ≈ 0.055 ≈ 5.5%

Therefore, the percentage growth rate, "r", is approximately 2.8% or 5.5%.