A forestry service uses a remote weather station powered by a solar battery system. During… — SAT

A forestry service uses a remote weather station powered by a solar battery system. During a period of heavy cloud cover, the remaining charge in the battery, $C$ , in watt-hours, can be modeled by the linear equation $C = 360 - 4.5h$ , where $h$ represents the number of hours since the cloud cover began. Based on the model, what is the best interpretation of the $h$ -intercept of the relationship?

The number of hours, which is 80, after the cloud cover began when the battery is completely discharged.Answer
B
The initial charge of the battery, which is 360 watt-hours, when the cloud cover began.
C
The rate, which is 4.5 watt-hours per hour, at which the battery charge decreases.
D
The number of hours, which is 90, after the cloud cover began when the battery is completely discharged.

Answer

The number of hours, which is 80, after the cloud cover began when the battery is completely discharged.

The

h

-intercept of the relationship is the value of

h

when

C = 0

. Substituting

0

for

C

in the equation

C = 360 - 4.5h

gives

0 = 360 - 4.5h

. Solving for

h

yields

4.5h = 360

, or

h = 80

. In context, this represents the number of hours after the cloud cover began when the battery is completely discharged.

Step-by-Step Solution

Identify the meaning of the

h

-intercept in the context of the linear equation.

The

h

-intercept occurs when the dependent variable,

C

(remaining charge), is equal to 0.

By definition, the horizontal intercept of a function occurs where the vertical coordinate is zero.

Substitute

C = 0

into the equation and solve for

h

0 = 360 - 4.5h \implies 4.5h = 360 \implies h = 80

This algebraic isolation finds the specific value of

h

when the charge is completely depleted.

Interpret the resulting value of

h

in the context of the problem.

h = 80

hours, the remaining battery charge is 0 watt-hours, meaning the battery is fully discharged.

Connecting the mathematical coordinate

(80, 0)

back to the units of hours and watt-hours provides the physical interpretation.

Key Concept

Interpreting Linear Relationships in Context

Question

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