School Roof Solar: Balancing the Changes

Interpret opposite quantities in a rooftop solar project.

Read the Energy Balance

The school is installing rooftop solar panels. In these energy records, energy entering the school's battery is positive and energy leaving it is negative. Energy is measured in kilowatt-hours (kWh). Ignore energy losses.

During one test interval, the panels send 7 kWh into the battery while school equipment draws 7 kWh from it. Which statement describes the battery's net energy change?

A. The net change is +14 kWh because both amounts count.B. The net change is +7 kWh because panels produce energy.C. The net change is −7 kWh because equipment uses energy.D. The net change is 0 kWh because equal opposite amounts balance.

Select every statement that must be true when two nonzero signed energy changes are opposites.

The battery monitor records a net change of zero during a test. One energy transfer is shown in the table.

Battery Test
Transfer	Signed energy (kWh)
Energy entering	?
Energy leaving	−2.75

What positive signed value belongs in the missing cell? kWh

Which description fits the equation +4.5 + (−4.5) = 0 in the battery's energy record?

A. The battery receives 4.5 kWh and supplies 4.5 kWh.B. The battery supplies 4.5 kWh and supplies 4.5 kWh more.C. The battery receives 4.5 kWh and receives 4.5 kWh more.D. The battery receives 4.5 kWh and supplies no energy.

During a demonstration, the panels send 3/4 kWh into the battery. A lamp and a fan draw energy from it. Which plan makes the battery's net energy change zero?

A. The lamp draws 1/2 kWh and the fan draws 1/2 kWh.B. The lamp draws 1/4 kWh and the fan draws 1/2 kWh.C. The lamp draws 3/4 kWh and the fan draws 1/4 kWh.D. The lamp draws 1/4 kWh and the fan draws 1/4 kWh.

Balance the Project Plans

Identify what is changing and check the units. Quantities cancel only when they describe equal amounts of the same quantity in opposite directions.

Adding equipment increases the mass on the roof; removing equipment decreases it. Which installation plan leaves the total mass on the roof unchanged?

Installation Plans
Plan	Mass added (kg)	Mass removed (kg)
Cedar	48	42
Birch	54	54
Elm	42	48
Maple	54	45

A. The Birch plan leaves the mass unchanged.B. The Maple plan leaves the mass unchanged.C. The Elm plan leaves the mass unchanged.D. The Cedar plan leaves the mass unchanged.

Each of four new panels adds 12 kg to the roof. The installer will remove old equipment so that the total mass on the roof does not change.

What signed mass change must the removal produce? kg

In a roof layout, eastward movement is positive and westward movement is negative. A mounting rail is shifted 5/8 meter west. Which movement would cancel that shift?

A. Shift the rail 3/8 meter east.B. Shift the rail 5/8 meter east.C. Shift the rail 8/5 meter east.D. Shift the rail 5/8 meter west.

Select every project record in which the two changes can combine to make zero. Consider both the amounts and what they measure.

10.

A design revision adds 2 1/2 m² of panel-covered area to the roof plan. The designers also remove an equal area of planned panels elsewhere. Increases in panel-covered area are positive.

What is the combined signed change in panel-covered area? m²

Check the Claims

Use opposite quantities to evaluate the project team's conclusions.

11.

The energy record contains +5 kWh and −4 kWh. Jordan says the changes cancel because one is positive and one is negative. Which correction is valid?

A. They cancel because both changes use kilowatt-hours.B. They do not cancel because neither entry is zero.C. They do not cancel because the magnitudes differ.D. They cancel because the two signs are different.

12.

During a battery test, the panels supply two equal energy inputs. School equipment draws 9 kWh in total. These are the only transfers, and the net energy change is zero.

How much energy does each input supply? kWh

13.

The battery's net energy change is zero during a test. Which additional record would prove that energy moved both into and out of the battery?

A. The roof held twelve panels throughout the test.B. The battery monitor stayed on throughout the test.C. The battery received 1.25 kWh during the test.D. The test began at the scheduled starting time.

14.

Two groups of rooftop panels each send 1.5 kWh into the battery. School equipment draws 3 kWh. These are the only transfers. Priya says, “The net change cannot be zero because no single positive entry is the opposite of −3.” Explain why the net change is zero even without a single +3 entry. Include an equation.