Question

The table below shows the number of new restaurants in a fast-food chain that opened during the years 1988 through 1992. Using an exponential model... write an equation for the curve of best fit, Year New Restaurants 1988 49 1989 81 1990 112 1991 150 1992 262

Answer 1

New restaurant growth: y =
`29.64*1.653^x`. Predicted 1993 openings: 462.

To find the equation for the curve of best fit, we can use an exponential model.

The general form of an exponential model is:

y = a x
`b^x`

where:

y is the dependent variable (in this case, the number of new restaurants)

x is the independent variable (in this case, the year)

a is the initial value (the number of new restaurants in 1988)

b is the growth factor (the rate at which the number of new restaurants increases each year)

We can use the data in the table to solve for a and b.

Let's substitute the data for 1988 into the equation:

49 = a x
`b^1`

We can also substitute the data for 1989:

81 = a x
`b^2`

Now we have two equations with two unknowns. We can solve for a and b using elimination. Dividing the second equation by the first equation, we get:

1.653 = b

Substituting this value of b back into the first equation, we get:

49 = a x
`1.653^1`

a = 29.64

Therefore, the equation for the curve of best fit is:

y = 29.64 x
`1.653^x`

This equation can be used to predict the number of new restaurants that will open in future years. For example, to predict the number of new restaurants that will open in 1993, we can substitute x = 6 into the equation:

y = 29.64 x
`1.653^6`

y = 462.33 ≈ 462

Therefore, we predict that approximately 462 new restaurants will open in 1993.