A homeowner wishes to insulate her attic with fiberglass insulation to conserve energy. The insulation comes in 40cm wide rolls that are cut to fit between the rafters in the attic. If the roof is 6m from peak to eave and the attic space is 2cm high at the peak, how long does each of the pieces of insulation need to be?

Question

A homeowner wishes to insulate her attic with fiberglass insulation to conserve energy. The insulation comes in 40cm wide rolls that are cut to fit between the rafters in the attic. If the roof is 6m from peak to eave and the attic space is 2cm high at the peak, how long does each of the pieces of insulation need to be?

Answer 1

To determine the length of each piece of insulation, we need to find the length of the diagonal distance from the peak to the eave of the attic. This can be calculated using the Pythagorean theorem, which relates the lengths of the sides of a right triangle.

In this case, the height of the attic space (2cm) represents one leg of the right triangle, and the diagonal distance (length of the insulation needed) is the hypotenuse. Given that the roof is 6m from peak to eave, we can calculate the length of the diagonal distance as follows:

Using the Pythagorean theorem:
Diagonal distance^2 = height^2 + roof length^2
Diagonal distance^2 = (2cm)^2 + (6m)^2

Now, we need to convert the height to meters to work with consistent units:
Height in meters = 2cm / 100 (since there are 100 centimeters in a meter)

Diagonal distance^2 = (0.02m)^2 + (6m)^2
Diagonal distance^2 = 0.0004m^2 + 36m^2
Diagonal distance^2 = 36.0004m^2

Finally, we can find the diagonal distance:
Diagonal distance = √36.0004m^2

Using a calculator or software, we find the diagonal distance to be approximately 6.00006667 meters.

Since the insulation comes in 40cm wide rolls, we need to divide the diagonal distance by the width of the roll to determine the number of rolls needed. Converting the width to meters:

Width of roll in meters = 40cm / 100

Rolls needed = Diagonal distance / Width of roll

Rolls needed = 6.00006667m / (40cm / 100)

Using a calculator or software, we find the number of rolls needed to be approximately 15.00016667 rolls.

Since we can't purchase fractional rolls, we would need to round up to the nearest whole number. Therefore, we would need 16 rolls of insulation.

To calculate the length of each piece of insulation:
Length of each piece = Length of attic space / Number of rolls needed

Length of each piece = 6m / 16

Using a calculator or software, we find the length of each piece of insulation to be approximately 0.375m or 375mm.