A businessman models the number of items (in thousands) that his company sold fr

Question

A businessman models the number of items (in thousands) that his company sold from 1998 through 2004 as N(x) = -0.1x^3 + x^2 - 3x + 4 and the average price per item (in dollars) as P(x) = 0.2x + 5, where x represents the number of years since 1998. Write a polynomial R(x) that can be used to model the total revenue for this company. A biologist has found that the number of branches on a certain rare tree in its first few years of life can be modeled by the polynomial b(x) = 4x^2 + x. The number of leaves on each branch can be modeled by the polynomial L(x) = 2x^3 + 3x^2 + x, where x is the number of years after the tree reaches a height of 6 feet. Write a polynomial describing the total number of leaves on the tree. The dimensions for a rectangular prism are x + 5 for the length, x + 1 for the width, and x for the height. What is the volume of the prism? (V = l wh) Mark runs a small toy store specializing in wooden toys. From 2000 through 2012. the number of toys Mark made can be modeled by N(x) = 0.7x^2 - 2x + 23, and the average cost to make each toy can be modeled by C(x) = -0.004x^2 - 0.08x + 25, where x is the number of years since 2000. Write a polynomial that can be used to model Mark's total cost for making the toys, T(x), for those years.

Explanation / Answer

Since Revenue ( say R(x)) = Quantitysold * Price , therefore, R(x) = N(x) *P(x) = ( -0.1x3 +x2 -3x+4)*( 0.2x+5) = -0.02x4 +0.2x3 -0.6x2+ 0.8x -0.5x3+5x2 -15x +20 or, R(x) = -0.02x4 -0.3x3+4.4x2 -14.2x +20. The total number of leaves on the rare tree, after it reaches a height of 6 ft. = no.of branches* no. of leaves per branch = b(x)* L(x) = (4x2+x)*(2x3 +3x2+x) = 8x5+12x4+4x3+ 2x4 +3x3+ x2 = 8x5+14x4+7x3+ x2. The volume (V) of the prism is given by V = lwh = (x+5)(x+1)(x)= x(x2+6x+5)= x3+6x2+5x. Total cost T(x) = Total no. of toys * cost per toy = N(x)*C(x) = (0.7x2-2x+23)*(-0.004x2-0.08x+25) = - 0.028x4+ 0.008x3-0.092x2-0.056x3+0.16x2-1.84x+17.5x2 -50x +575 = - 0.028x4- 0.048x3-0.092x2+17.568x -51.84x+575

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