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Chapter 49: Chapter 49

Problem 854

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The base of a right prism, as shown in the figure, is an equilateral triangle, each of whose sides measures 4 units. The altitude of the prism measures 5 units. Find the volume of the prism. [Leave answer in radical form.]

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The volume of the right prism with an equilateral triangular base of side 4 units and altitude of 5 units is \(20\sqrt{3}\) cubic units.
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Find the area of the equilateral triangle base

To find the area of an equilateral triangle with side length 'a', we need to use the formula: \[Area = \frac{\sqrt{3}}{4}a^2\] In our case, the side length (a) is 4 units. Now, we will calculate the area of the triangle using the given formula: \[Area = \frac{\sqrt{3}}{4}(4^2)\]

Step 2: Simplify the expression of the area

After substituting the value of a, now we simplify the expression: \[Area = \frac{\sqrt{3}}{4}(16)\] \[Area = 4\sqrt{3}\]

Step 3: Find the volume of the prism

Now we have the area of the base of the prism (4√3) and the altitude of the prism (5). The formula for finding the volume of a prism is: \[Volume = (Area \space of \space base) \times h\] Here, h stands for the altitude (height) of the prism. Substituting the area of the base and altitude in the equation, we get: \[Volume = (4\sqrt{3}) \times 5\]

Step 4: Simplify the expression and write the final answer in radical form

Finally, we simplify the expression to get the volume of the prism: \[Volume = 20\sqrt{3}\] So, the volume of the right prism with an equilateral triangular base of side 4 units and altitude of 5 units is \(20\sqrt{3}\) cubic units.

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Most popular questions from this chapter

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How many cubic feet are contained in a packing case which is a rectangular solid 4 ft. long, \(3 \mathrm{ft}\). wide and \(3(1 / 2) \mathrm{ft}\). high?
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A regular pyramid has a pentagon for its base. (a) Show that the area of the base is given by the formula \(\beta=(5 / 4) \mathrm{e}^{2} \tan 54^{\circ}\) where \(e=\) an edge of the base. (a) If the slant height of the pyramid makes an angle of \(54^{\circ}\) with the altitude of the pyramid, show that the altitude is given by the formula \(\mathrm{h}=(\mathrm{e} / 2)\). (b) Using the formulas of part a and \(b\), write a formula for the volume of the pyramid in terms of the base edge e.
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