ALGEBRA 2- Exam 2 An Egyptian pharaoh has ordered the construction of four squar

Question

ALGEBRA 2- Exam 2 An Egyptian pharaoh has ordered the construction of four square pyramids. The will be the tallest, while the other three pyramids will each be half its height, must have a height to base length ratio of 2:3. The pharaoh has allocated a of 30 million cubic feet of stone for construction. The mairn ht. All four ALGL Exan Base Stuc notation to 3 decimal places. Hint: Rece Bh. where B is the area of the base and h is the heigh largest possible values for the base length and height, to the the volume of each pyramid, in se volume of a pyramid is V- 29. Find the base length: height: volume: base length:- height: volume: FILL IN YOUR NAME AND THE OTHER REQUIRED INFORMATION ON EACH P OF THE EXAM AND MAIL THE EXAM TOTHE COU

Explanation / Answer

The volume of the pyramid is V =(1/3 )Bh, where B is the area of the base and h is the height. Here, the pyramids have square bases so the area of the base equals the square of the length of the base. Let the height of each of the smaller pyramids be h. Since the ratio of height to the pyramid to the length of its base is 2:3, the length of the base of the smaller pyramid is 3h/2. Thus, the volume of the smaller pyramid is 1/3 (3h/2)(3h/2)h = 3h3/4 so that the aggregate volume of the 3 smaller pyramids is 3*3h3/4 = 9h3/4.

Further, the main pyramid has a height 2h and the length of its base is (3/2)(2h) = 3h. Thus, the volume of the main pyramid is 1/3( 3h)(3h)(2h) = 6h3. Now, since the maximum aggregate volume (V) of the 4 pyramids is 30 million cubic feet, we have 9h3/4 +6h3 30 or, 33 h3/4 30 or, h3 30*4/33 or,h3 40/11. Then h (40/11)1/3 or, h 1.53776192 million feet or, h 1537761.92 feet so that 2h 3075523.84 feet. Also, 3h/2 2306642.88 feet and 3h 4613285.76.

Thus, the maximum height and base length for the smaller pyramid is 1537761.92 feet and 2306642.88 feet respectively. Also, the maximum height and base length for the main pyramid are 3075523.84 feet and 4613285.76 feet respectively.

Note:

We have rounded off to 2 decimal places.

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