Exercise 152 Page 703

a We want to find how much fabric does the following small tent use.

The tent is a triangular prism and the fabric is used for the surface area. Therefore, we want to find the surface area of the prism using the given diagram. The surface area of a prism can be found using the following formula. S = 2B + Ph

Here, B is the base's area, P is the base's perimeter, and h is the height of the prism. The base is an isosceles triangle with one side measuring 2 feet and the height falling onto that side is 3 feet. Area of a triangle is half the product of a side and the height falling onto that side. Let's find the area the base using this formula! B = 1/2(2)(3) = 3 The base is 3 square feet. Next, let's find the base's perimeter. We know one that one of the sides is 2 feet. Since the prism's base is an isosceles triangle with base of 2 feet, so the height falling onto that side also bisects this side. Let's draw this situation on a diagram.

The height creates two congruent right triangles. Their hypotenuses are the two missing sides of the larger triangle. To find them, we can use the Pythagorean Theorem. a^2 + b^2 = c^2 Here, a and b are the legs of a right triangle and c is the hypotenuse. In our case, the legs are 1 feet and 3 feet. Let's substitute 1 for a and 3 for b in the above formula and solve the resulting equation for the c, the hypotenuse.

a^2 + b^2 = c^2

SubstituteII

a= 1, b= 3

1^2 + 3^3 = c^2

▼

Solve for c^2

CalcPow

Calculate power

1+ 9 = c^2

AddTerms

Add terms

10 = c^2

RearrangeEqn

Rearrange equation

c^2 = 10

SqrtEqn

sqrt(LHS)=sqrt(RHS)

c = sqrt(10)

UseCalc

Use a calculator

c = 3.162277...

RoundDec

Round to 2 decimal place(s)

c ≈ 3.16

The hypotenuse is about 3.16 feet. Note that we only care about the principal root, since the hypotenuse is positive. The perimeter of the prism's base has two such hypotenuses as the sides and the third side is 2 feet long. Let's use this fact to find the perimeter P. P ≈ 2(3.16) + 2 = 8.32 The perimeter is about 8.32 feet. From the original diagram we also have that the prism's height is 3 feet. We also found that the base's area is 3 square feet. Let's substitute 3 for B, about 8.32 for P, and 3 for h in the formula for the surface area and simplify.

S = 2B + Ph

SubstituteValues

Substitute values

S ≈ 2( 3) + 8.32( 3)

Multiply

S ≈ 6+ 24.96

AddTerms

Add terms

S ≈ 30.96

RoundInt

Round to nearest integer

S ≈ 31

The surface area of the prism is about 31 square feet, so we need abut 31 square feet of fabric for the tent.

b In Part B, we found that the small tent is a triangular prism. Let's recall the formula for the volume of a prism.

V = Bh Here, B is the area of the prism's base and h is the prism's height. In Part B, we also found that the prism's base is 3 square feet and it's height is 3 feet. Let's substitute 3 for B and 3 for h in the above formula and simplify to find the volume of the small tent.

V = Bh

SubstituteII

B= 3, h= 3

V = 3( 3)

Multiply

V = 9

The small tent's volume is 9 cubic feet.

c We know that the volume of the full-sized tent is 72 cubic feet. In Part B, we found that the volume of the miniature is 9 cubic feet. The model is similar to the full-sized tent, so we can find the volume scale factor between the two by dividing the volume of the full-sized tent by the volume of the miniature. Let's do it!

Volume scale factor = 72/9 = 8 Next, we will find the linear scale factor between the two tents. The volume scale factor is the cube of the linear scale factor. Therefore, we can find the linear scale factor by taking the cube root of the volume scale factor.

Linear scale factor = sqrt(Volume scale factor)

Substitute

Volume scale factor= 8

Linear scale factor = sqrt(8)

CalcRoot

Calculate root

Linear scale factor = 2

The linear scale factor is 2. This scale factor tells us that all the lengths of the full-sized model are 2 times as long as the lengths of the miniature. Therefore, since the miniature is 3 feet high, the full-sized model is 2( 3) = 6 feet high.

d We want to find how much fabric the full-sized model uses. In Part A, we found that the miniature uses about 31 square feet of fabric. In Part C, we found that the linear scale factor between the miniature and the full-sized tent is 2.

Linear scale factor = 2

The area scale factor is a square of the linear scale factor. Area scale factor = ( 2)^2 ⇓ Area scale factor = 4 To get the amount of fabric we need for the full-sized tent we can multiply the amount of fabric we need for the miniature by the area scale factor. Let's do it! 4( 31) = 124 For the full-sized tent we need about 124 square feet of fabric.

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Exercise 152 Page 703

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