A residential solar power system stores electricity in a battery. The total energy stored … — SAT

A residential solar power system stores electricity in a battery. The total energy stored in the battery, $E$ , in kilowatt-hours (kWh), is modeled by a linear function of the time $t$ , in hours, since sunrise. After $4$ hours of sunlight, the energy stored is $18$ kWh. After $7$ hours of sunlight, the energy stored is $27$ kWh. Which of the following is the best interpretation of the slope of the graph of this function in the $tE$ -plane?

The energy stored in the battery increases by $3$ kilowatt-hours per hour of sunlight.Answer
B
The initial energy stored in the battery at sunrise is $6$ kilowatt-hours.
C
It takes $3$ hours of sunlight to increase the energy stored in the battery by $1$ kilowatt-hour.
D
The energy stored in the battery increases by $9$ kilowatt-hours per hour of sunlight.

Answer

The energy stored in the battery increases by

3

kilowatt-hours per hour of sunlight.

To find the slope of the linear relationship, we use the formula for the rate of change:

\text{slope} = \frac{\Delta E}{\Delta t}

. Using the two points

(4, 18)

and

(7, 27)

, the slope is

\frac{27 - 18}{7 - 4} = \frac{9}{3} = 3

kilowatt-hours per hour. In the context of the problem, the slope represents the rate at which the stored energy increases per hour of sunlight. Therefore, the energy stored in the battery increases by

3

kilowatt-hours per hour of sunlight.

Step-by-Step Solution

Identify the coordinates

(t, E)

from the given values.

The two data points are

(4, 18)

and

(7, 27)

These points represent the time in hours and the corresponding energy stored in the battery.

Calculate the slope of the linear relationship using the slope formula.

The slope is

\frac{27 - 18}{7 - 4} = \frac{9}{3} = 3

The slope of a linear relationship represents the constant rate of change of the dependent variable (

E

) per unit change of the independent variable (

t

Interpret the meaning of the calculated slope in the context of the problem.

A slope of

3

means that the energy stored in the battery,

E

, increases by

3

kilowatt-hours (kWh) for every

1

hour of sunlight,

t

The units of the slope are units of

E

divided by units of

t

, which is kilowatt-hours per hour.

Key Concept

Interpreting the rate of change (slope) of a linear relationship in context.

Estimated Time:1m 30s

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