Question

An exponential function in the form y = ab^x goes through the points (3, 10.125) and (6, 34.2). Find a to the

nearest integer and b to the nearest tenth, then find f (10) to the nearest integer.

Answer 1

`f(10) = 173`

Explanation:

Given

Exponential Function

`(x_1,y_1) = (3,10.125)`

`(x_2,y_2) = (6,34.2)`

Required

Determine f(10)

We have that

`y = ab^x`

First, we need to solve for the values of a and b

For
`(x_1,y_1) = (3,10.125)`

`10.125 = ab^3` --- (1)

For
`(x_2,y_2) = (6,34.2)`

`34.2 = ab^6` ---- (2)

Divide (2) by (1)

`(34.2)/(10.125) = (ab^6)/(ab^3)`

`(34.2)/(10.125) = (b^6)/(b^3)`

`3.38= b^(6-3)`

`3.38= b^(3)`

Take cube root of both sides

`b = \sqrt[3]{3.38}`

`b = 1.5`

Substitute 1.5 for b in
`10.125 = ab^3`

`10.125 = a * 1.5^3`

`10.125 = a * 3.375`

Solve for a

`a = (10.125)/(3.375)`

`a = 3`

To solve for f(10).

This implies that x = 10

So, we have:

`y = ab^x` which becomes

`y = 3 * 1.5^{10`

`y = 3 * 57.6650390625`

`y = 172.995117188`

`y = 173` -- approximated

Hence:

`f(10) = 173`