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 =
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.
(b) Find the percentage growth rate, r. (Round your answer to the nearest percent.)
r = %
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.
Been there, done that:
http://www.jiskha.com/display.cgi?id=1440890314
To find the base, a, in the equation f(t) = Q₀aᵗ, we can use the given information that f(7) = 75.94 and f(9) = 91.89.
For part (a), we have the equation 75.94 = Q₀a⁷. Similarly, for part (b), we have the equation 91.89 = Q₀a⁹.
To solve for a, we need to isolate it in one of the equations. We can divide the second equation by the first equation to eliminate Q₀:
(91.89/75.94) = (Q₀a⁹)/(Q₀a⁷)
1.2092 ≈ a²
Now we can take the square root of both sides:
√(1.2092) ≈ a
a ≈ 1.10 (rounded to two decimal places)
So the base, a, is approximately 1.10.
For part (b), to find the percentage growth rate, r, we can substitute the value of a into either of the original equations. Let's use the equation 75.94 = Q₀a⁷:
75.94 = Q₀(1.10)⁷
To solve for Q₀, we divide both sides by (1.10)⁷:
Q₀ ≈ 75.94 / (1.10)⁷
Q₀ ≈ 52.91 (rounded to two decimal places)
Now we can substitute the values of Q₀ and a into the second equation, 91.89 = Q₀(1 + r)⁹:
91.89 = 52.91(1 + r)⁹
To solve for r, we divide both sides by 52.91 and take the ninth root to isolate (1 + r):
(91.89 / 52.91)^(1/9) ≈ 1 + r
Now subtract 1 from both sides to find the value of r:
(91.89 / 52.91)^(1/9) - 1 ≈ r
Calculating this expression, we find that r ≈ 0.078 (rounded to three decimal places).
To convert this to a percentage, we multiply by 100:
r ≈ 0.078 * 100
r ≈ 7.8%
So, the percentage growth rate, r, is approximately 7.8%.