Question

Hi I rly rly need help with this. I have a test tomorrow lol

Answer 1

The equation that best models the given data is:

f(x) = 28x + 126.5

How to find the equation that best models the given data

To find the equation that best models the given data, use linear regression to determine the relationship between the length (x) and the area (f(x)).

Using the provided data points:

Length (x): 1, 2, 3, 4

Area (f(x)): 144, 179, 214, 249

We can formulate the equation in the form of:

f(x) = mx + b

where m represents the slope and b represents the y-intercept.

Step 1: Calculate the means of x and f(x):

Mean of x (
`\bar{x}`) = (1 + 2 + 3 + 4) / 4 = 2.5

Mean of f(x) (
`\bar{f}`) = (144 + 179 + 214 + 249) / 4 = 196.5

Step 2: Calculate the deviations from the means for x and f(x):

Deviation of x (x -
`\bar{x}`):

-1.5, -0.5, 0.5, 1.5

Deviation of f(x) (f(x) -
`\bar{f}`):

-52.5, -17.5, 17.5, 52.5

Step 3: Calculate the sum of the products of the deviations:

Sum of (x -
`\bar{x}`)(f(x) -
`\bar{f}`) = (-1.5 * -52.5) + (-0.5 * -17.5) + (0.5 * 17.5) + (1.5 * 52.5) = 140

Step 4: Calculate the sum of squared deviations of x:

`\sum (x - \bar{x})^2 = (-1.5)^2 + (-0.5)^2 + (0.5)^2 + (1.5)^2 = 5`

Step 5: Calculate the slope (m):

`m = \sum (x - \bar{x})(f(x) - \bar{f}) / \sum (x - \bar{x})^2`

m = 140 / 5 = 28

Step 6: Calculate the y-intercept (b):

b =
`\bar{f}` - m *
`\bar{x}`

b = 196.5 - 28 * 2.5 = 126.5

Therefore, the equation that best models the given data is:

f(x) = 28x + 126.5