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Question 1099545: An amusement park owner wants to add a new wilderness water ride to his park. The ride includes a mountain that is in the shape of a square based pyramid. Before building the attraction, engineers must build and test a scale model.
(a) If the length of the sides of the base of the scale model of the pyramid is xcm and the height of the pyramid is 9cm less than the length of the sides of the base, write down an equation for the volume of the pyramid in terms of x.
You may assume that the volume of a square based pyramid is
V = 1Bh, where B is the area of the base and h is the height of the pyramid.
(b) If the volume of the model of the pyramid is 6300cm3, show that
x^3-9x^2-18900=0
(d) Fully factorise p(x) and hence explain why the base of the model pyramid has sides of length 30cm.
(e) If the model of the pyramid is built to a scale of 4:250, determine the height of the pyramid to be built
in the park.
Answer by htmentor(1343)   (Show Source):
You can put this solution on YOUR website! The volume of the pyramid is V = (1/3)Bh. The area of the base, B is equal to x^2.
And since the height is 9 cm less than the length of the sides, h = x - 9
Therefore V in terms of x is:
a) V(x) = (1/3)x^2*(x-9)
b) (1/3)x^2*(x-9) = (1/3)x^3 - 3x^2 = 6300
Multiply through by 3:
x^3 - 9x^2 - 18900 = 0
c) We need to show that the length of the base, x = 30 cm
That implies that x - 30 must be a factor.
Dividing x^3 - 9x^2 - 18900 by x - 30 gives x^2 + 21x + 630, which cannot be factored any further
The factorization is (x-30)(x^2 + 21x + 630)
d) For the scale model h = x - 9 -> h = 30 - 9 = 21
If all dimensions are scaled up by the factor 250/4, the new height will be 250/4*21 = 1312.5 cm
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