A/P/V Question 3

Question

Find the volume of a rectangular prism that has a length of 20 cm, a width of 14 cm and a height of 6 cm. Give your answer to 1 decimal place if necessary. 

 

 

Find the total surface area AND volume of a square based pyramid with a base side length of 80 cm and an overall height of 150 cm.

 

A square based pyramid sits on a floor. Its sides are to be painted. Each side of the base is 1.5 m long and the distance along an edge from the vertex (peak) to a corner of the base is 2.0 m. What is the area of the pyramid to be painted? 

Answer 1

First problem

The volume of a rectangular prism with length L, Width W and height h is V = L * W * H.

With the dimensions given the volume is 1680 cm³

 

Second problem

The volume of a square prism with side s of the square and height h is V = s² * h /3

With the values given, V = 320000 cm³

 

There are 4 similar triangles that make the other sides of the pyramid.

For one these triangles, the altitude is the hypotenuse of a right triangle made

from height of the pyramid and half the side of a square.

Let a be the value of this altitude. a = √(h² + s²/4) = √(4 * h² + s²)/2

The base of the triangle is the same as the length of the base.

The are of the triangle is A_T = (1/2) * s * √(4 * h² + s²)/2

                                         = s * √(4 * h² + s²)/4

 

The area of the square base is A_S = s²

 

The total surface area is A = A_S + 4 * A_T

                                       = s² + s * √(4 * h² + s²)

With the values given A = 6400 + 80√(96400)

                                   = 6400 + 1600√(241) ≈ 31238.68 cm²

 

Third problem
Since the base is sitting of the floor assume it will not be painted.

 

The other 4 sides are similar triangles with the base b the side of the square.

b = 1.5 m

 

Let h be the altitude of one triangles and s the side of the triangle from

vertex to corner.  With half the size of the base this is a right triangle,

a = √(s² - (b/2)²) = √(4 * s² - b²)/2

The area of this triangle is A_T = (1/2) b * a = b * √(4 * s² - b²)/4

 

The area to be painted are the four triangles that make up the perimeter

A = 4A_T = b * √(4 * s² - b²)

 

With the values given, A = 1.5 + √(13.75) ≈ 5.56 m²

 

Answer 2

Amber,

This prism has the shape of a box.

To find the volume, you must take the length times the width times the height.

For this prism, the volume is (20 cm)(14 cm)(6 cm) = 1680 cm³.

 

For the pyramid, there is a volume formula as well: V = (1/3) Bh, where B is the area of the square base and h is the height.

 

For the problem you were given, B = (80 cm)² = 6400 cm² and h = 150 cm, so the volume is

 

V = (1/3)(6400 cm²)(150 cm) = 320,000 cm³.

 

The surface area is a little harder. The pyramid has five sides, including the square base and the four lateral sides what are triangles of the same size and shape, so they have the same area.

 

The formula for the surface area in this problem is then

 

S = B + 4T,

 

where B is the area of the base above and T is the area of a lateral triangular face. You already know B, so we must concentrate on finding T.

 

The area of a triangle is ½(base)(height). You were given the base is 80 cm, but the height is measured along the face and is not the same as the pyramid height. To find it, you need to form a right triangle as follows. The base is half of the base of the pyramid. It extends from the center of the base to the midpoint of a side of the base. This will be 40 cm. The height is the height of the pyramid, or 150 cm. The hypotenuse is the height of the triangle you are trying to find the area for.

 

Using the Pythagorean Theorem, the height of this triangle is √(40² + 150²) = √(1600 + 22500) = √24100, so

 

T = ½(80 cm)(√24100 cm) ≈ 6209.7 cm².

 

To finish the problem, you find the total surface area:

 

S = B + 4T = 6400 + 4(6209.7) ≈ 31,238.7 cm².

 

For the last question, I will assume that the base is not to be painted. The area to be painted is then 

 

A = 4T,

 

where T is the area of one of the four triangular sides. Each triangle has base b =1.5 m and the other two sides have length 2.0 m (the triangle is isosceles because two sides are equal). You still need the height, so draw the height h and find it using the Pythagorean Theorem again:

 

h² + (1.5/2)² = 2.0²

 

h² + 0.5625 = 4

 

h² = 4 − 0.5625 = 3.4375

 

h = √3.4375 ≈ 1.854 m.

 

The area T is now ½(1.5 m)(1.854 m) = 1.3905 m², and the total area to be painted is

 

A = 4T ≈ 5.562 m².

 

(If the last answer must be rounded to one decimal place as well, it should be given as 5.6 m².)

 

I am sorry for the long explanation, but I tend to use more words than most tutors when I explain, just to help you understand better and keep you focused on the solution. Also, I am unable to draw pictures here which is another reason.