A scientist measures the density of a core sample of ice as a function of depth. The densi… — SAT

A scientist measures the density of a core sample of ice as a function of depth. The density $d$ , in grams per cubic centimeter ( $\text{g/cm}^3$ ), of the ice at a depth of $x$ meters below the glacier surface is modeled by the equation $d = 0.0004x + 0.917$ . According to the model, what is the depth, in meters, at which the ice density is $0.935\text{ g/cm}^3$ ?

Answer: 45 meters

Answer

The depth is 45 meters.

To find the depth at which the density is

0.935\text{ g/cm}^3

, substitute

0.935

for the density

d

in the model equation, yielding

0.935 = 0.0004x + 0.917

. Subtracting

0.917

from both sides gives

0.018 = 0.0004x

. Dividing both sides by

0.0004

results in

x = 45

. Therefore, the depth is 45 meters.

Step-by-Step Solution

Substitute the target density of

0.935\text{ g/cm}^3

for

d

in the given model equation.

0.935 = 0.0004x + 0.917

This sets up the linear equation to solve for the corresponding depth

x

Subtract

0.917

from both sides of the equation.

0.018 = 0.0004x

This isolates the term containing the variable

x

on one side of the equation.

Divide both sides of the equation by

0.0004

to find the value of

x

x = 45

Dividing

0.018

0.0004

solves for the depth

x

in meters.

Key Concept

Solving linear relationships in context for the independent variable given a value of the dependent variable.

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