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volume of frustum of a pyramid

by Mateta Lubinda
(Zambia)











































A hole is to be dug in the form of a frustum of a pyramid.

The top is to be a square of side 6.40 m and the bottom a square of 3.60 m.

The depth of the hole is to be 4.00 m.

Calculate the following:


1. The volume of the earth to be removed.

2. If the hole is now filled with concrete to a depth of 2.00 m, find the amount of concrete required.

Comments for volume of frustum of a pyramid

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Mar 29, 2014
volume of a frustum
by: Staff


Answer

Part I

The frustum of a square pyramid is a square pyramid with the top truncated (chopped off).

1. Calculate the volume of the earth to be removed.


Hole dug in the form of a frustum of a square pyramid




Volume of Frustum (the same formula applies to both the frustum of a pyramid and the frustum of a cone)

V = (h / 3) * [A1 + A2 + √(A1 * A2)]

V = Vfrustum = volume of frustum in meters3

h = height of frustum in meters

A1 = area of upper base in meters2

A2 = area of lower base in meters2



Volume of frustum of a square pyramid






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Mar 29, 2014
volume of a frustum
by: Staff


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Part II


h = 4.00 meters

A1 = 6.40 meters * 6.40 meters = 40.96 meters2

A2 = 3.60 meters * 3.60 meters = 12.96 meters2

V = (4.00 / 3) * [40.96 + 12.96 + √(40.96 * 12.96)]

V = (4.00 / 3) * [53.92 + √(530.8416)]

V ≈ (4.00 / 3) * [53.92 + 23.04]

V ≈ (4.00 / 3) * [76.96]

V ≈ 102.6133333

V ≈ 102.6 meters3


Volume of earth removed for frustum




Final Answer, part I:

the volume of earth to be removed ≈ 102.6 meters3



2. Calculate the concrete required if the hole is filled with concrete to a depth of 2.00 m .



Hole dug in the form of a frustum of a pyramid, and then partially filled with concrete








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Mar 29, 2014
volume of a frustum
by: Staff


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Part III


Solving for the upper side of the concrete base, x

The upper side of the concrete base is located "exactly" in the one-half the distance between the top and bottom of the hole.

Since each side of the hole has the shape of a trapezoid, the upper concrete base is located in the center of the trapezoid, half way between the two bases. This is a line segment called the median (also called the midline, or midsegment).

called the (note that the slant height of the trapezoid is not the same as the depth of the hole)



The upper base of the concrete is the median of the trapezoid





The length of the median line segment, x, is the average of the upper and lower base lengths.

x = (upper base + lower base) / 2

x = (6.4 + 3.6) / 2

x = (10) / 2

x = 5.0 m

The volume of the concrete required if the hole is filled with concrete to a depth of 2.00 m is calculated using the volume formula


V = (h / 3) * [A1 + A2 + √(A1 * A2)]

V = Vfrustum = volume of frustum in meters3

h = depth of concrete in meters

A1 = area of upper base of the concrete in meters2

A2 = area of lower base of the concrete in meters2

h = 2.00 meters

A1 = 5.0 meters * 5.0 meters = 25.0 meters2

A2 = 3.60 meters * 3.60 meters = 12.96 meters2

V = (2.00 / 3) * [25.0 + 12.96 + √(25.0 * 12.96)]

V = (2.00 / 3) * [37.96 + √(324)]

V ≈ (2.00 / 3) * [37.96 + 18]

V ≈ (2.00 / 3) * [55.96]

V ≈ 37.30666667

V ≈ 37.3 meters3




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Mar 29, 2014
volume of a frustum
by: Staff

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Part IV


the volume of the concrete poured into the bottom 2 m of the frustum





Final Answer, part II:

the volume of concrete ≈ 74.6 meters3



frustum volume:  earth removed and concrete poured into the bottom 2 m of the frustum








Apr 02, 2014
volume of a frustum
by: Staff

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Part V

A final note.

The calculation of the upper base of the concrete was easy because it is the median of the trapezoid.

As shown above:

The length of the median line segment, x, is the average of the upper and lower base lengths.

x = (upper base + lower base) / 2

x = (6.4 + 3.6) / 2

x = (10) / 2

x = 5.0 m


However, suppose the upper base of the concrete was not the median.

Suppose it was in some other position.

You can still calculate the length of the concrete upper base using a quadratic equation.

For comparison purposes, I'm going to calculate the length of the upper base for h/2 (the median) using a quadratic equation. However, this method can be used when the upper base is in any position, not just when it is the median.

V₁ (upper frustum) + V₂ (concrete base) = total volume of frustum = 102.6133333 (calculated in part I)

V₁ = (2.00 / 3) * [6.40² + x² + √(6.40² * x²)]

V₂ = (2.00 / 3) * [x² + 3.60² + √( x² * 3.60²)]


(2.00 / 3) * [6.40² + x² + √(6.40² * x²)] + (2.00 / 3) * [x² + 3.60² + √( x² * 3.60²)] = 102.6133333

(2.00 / 3) * [6.40² + x² + 6.40x)] + (2.00 / 3) * [x² + 3.60² + 3.6x] = 102.6133333

(2.00 / 3) * {[6.40² + x² + 6.40x)] + [x² + 3.60² + 3.6x]} = 102.6133333

(2.00 / 3) * {[6.40² + 3.60²+ x² + x² + 6.40x + 3.6x)] +]} = 102.6133333

(2.00 / 3) * { x² + x² + 6.40x + 3.6x + 6.40² + 3.60²} = 102.6133333

(2.00 / 3) * { 2x² + 10x + 53.92} = 102.6133333

(3 / 2.00) * (2.00 / 3) * { 2x² + 10x + 53.92} = (3 / 2.00) * 102.6133333

2x² + 10x + 53.92 - 153.91999995 = 153.91999995 - 153.91999995

2x² + 10x - 100 = 0

This is a quadratic equation which can be solved using the quadratic formula.

When this is done, the solution is:

x ≈ {-10, 5}

since x must be a positive value, the answer is x ≈ 5 meters

this is exactly the same value we calculated for the median by averaging the upper and lower base lengths






Thanks for writing.

Staff
www.solving-math-problems.com



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