A linear model is used to estimate the height of a plant, in centimeters, based on the num… — SAT

A linear model is used to estimate the height of a plant, in centimeters, based on the number of weeks since it was planted. According to the model, the plant's height is $5$ centimeters at week $2$ , and its height is $8$ centimeters at week $8$ . Which of the following is the predicted height of the plant, in centimeters, at week $16$ ?

A
$14$
B
$29$
$12$ Answer
D
$47$

Answer

The predicted height of the plant at week 16 is 12 centimeters.

The correct answer is 12. To find the predicted height, we determine the linear model equation. The slope is calculated as

m = \frac{8 - 5}{8 - 2} = \frac{3}{6} = 0.5

. Substituting the point

(2, 5)

into the slope-intercept form

y = mx + b

gives

5 = 0.5(2) + b

, which simplifies to

b = 4

. The linear model is

y = 0.5x + 4

. Substituting

16

for

x

yields

y = 0.5(16) + 4 = 8 + 4 = 12

Step-by-Step Solution

Identify two data points from the given information to represent the linear relationship.

The coordinates are

(2, 5)

and

(8, 8)

where the x-coordinate represents the week and the y-coordinate represents the height of the plant in centimeters.

A linear relationship is uniquely determined by two points, which allows us to find the slope and equation of the line.

Calculate the slope of the line, which represents the constant growth rate of the plant.

The slope

m

\frac{8 - 5}{8 - 2} = \frac{3}{6} = 0.5

centimeters per week.

The slope represents the constant rate of change of the plant's height per week.

Determine the y-intercept of the linear model using one of the coordinates.

Using the point

(2, 5)

and the equation

y = mx + b

, we get

5 = 0.5(2) + b

, which simplifies to

b = 4

Finding the y-intercept allows us to write the complete equation for the linear function representing the plant's growth.

Substitute the target week into the linear function to predict the height of the plant.

For week 16,

y = 0.5(16) + 4 = 8 + 4 = 12

centimeters.

Evaluating the function at

x = 16

yields the predicted height of the plant at that specific time.

Key Concept

Finding and evaluating linear functions using two given coordinate points.

Estimated Time:1m 30s

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