Question

Write an exponential model given the two points (6,150) and (7,250).

Answer 1

The exponential model representing the given points
`(6, 150) and (7, 250) is \(y = 100 * 2^x\).`

Step-by-step explanation:

To construct an exponential model, we can use the general form
`\(y = a * b^x\),` where
`\(a\)` is the initial value or y-intercept,
`\(b\)` is the base of the exponential function, and
`\(x\)` is the independent variable. Given the points (6, 150) and (7, 250), we can use the values to determine the specific coefficients of the model.

Starting with the point (6, 150), we substitute
`\(x = 6\) and \(y = 150\)` into the general form:

`\[150 = a * b^6\]`

Similarly, for the point (7, 250):

`\[250 = a * b^7\]`

Dividing the second equation by the first eliminates
`\(a\):`

`\[(250)/(150) = (a * b^7)/(a * b^6)\]`

Simplifying, we get:

`\[(5)/(3) = b\]`

Now, substitute
`\(b = (5)/(3)\)` into the first equation to find
`\(a\):`

`\[150 = a * \left((5)/(3)\right)^6\]`

Solving for
`\(a\)` , we get:

`\[a = 100\]`

Therefore, the exponential model is
`\(y = 100 * \left((5)/(3)\right)^x\),` which can be simplified to
`\(y = 100 * 2^x\).` This model accurately represents the exponential growth exhibited by the given data points.

Answer 2

The exponential model with the given points (6,150) and (7,250) is f(x) =
`(4374)/(635)((5)/(3) )^x`.

The general form of an exponential model is expressed as:

f(x) =
`ab^x`

Where a is the initial value and b is the base of the exponential function:

Given that the exponential model has the points (6,150) and (7,250).

f(6) = ab⁶ = 150

f(7) = ab⁷ = 250

Now, we divide the second equation by the first equation:

(ab⁷)/(ab⁶) = 250/150

b = 250/150

b = 5/3

Now, substitute b = 5/3 into the first equation and solve for a:

ab⁶ = 150

a(5/3)⁶ = 150

a = 4374/625

Now, plug the value of a and b into the exponential model formula:

f(x) =
`ab^x`

f(x) =
`(4374)/(635)((5)/(3) )^x`

Therefore, the equation of the exponential m odel is f(x) =
`(4374)/(635)((5)/(3) )^x`.