A solar power system stores electricity in a battery at a constant rate during the day. Th… — SAT

A solar power system stores electricity in a battery at a constant rate during the day. The total amount of energy in the battery, in kilowatt-hours (kWh), is a linear function of the number of hours after sunrise. Exactly $3$ hours after sunrise, the battery contains $14$ kWh of energy, and exactly $7$ hours after sunrise, it contains $26$ kWh of energy. If the battery's maximum capacity is $35$ kWh, how many hours after sunrise will it take for the battery to be fully charged?

Answer: 10 hours

Answer

To find the number of hours after sunrise it takes to fully charge the battery to 35 kWh, we first determine the constant rate of energy storage. The rate of change is given by the difference in energy divided by the difference in hours:

\frac{26 - 14}{7 - 3} = \frac{12}{4} = 3

kWh per hour. The relationship between energy

E

and hours

h

can be written as

E = 3h + b

. Substituting the known point

(3, 14)

gives

14 = 3(3) + b

, which simplifies to

b = 5

. Thus, the linear equation representing this relationship is

E = 3h + 5

. To find the time when the battery is fully charged to

35

kWh, set

E = 35

in the equation:

35 = 3h + 5

. Subtracting 5 from both sides yields

30 = 3h

, and dividing by 3 gives

h = 10

hours.

Step-by-Step Solution

Calculate the rate of energy storage per hour.

3 kWh per hour

Since the energy increases at a constant rate, the rate is the change in energy divided by the change in time:

\frac{26 - 14}{7 - 3} = 3

Determine the initial energy stored at sunrise.

5 kWh

Using the rate of 3 kWh per hour and the fact that the battery had 14 kWh at 3 hours, we subtract the energy gained in those 3 hours:

14 - 3(3) = 5

Set up the linear equation and solve for the time when the battery reaches 35 kWh.

10 hours

The linear relationship is

E = 3h + 5

. Setting

E = 35

gives

35 = 3h + 5

. Subtracting 5 from both sides gives

30 = 3h

, which simplifies to

h = 10

Key Concept

Linear Equations in Two Variables

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