91Ó°ÊÓ

The volume, \(V\), in cubic centimeters, of a collection of open-topped boxes can be modelled by \(V(x)=4 x^{3}-220 x^{2}+2800 x\) where \(x\) is the height, in centimeters, of each box. a) Use technology to graph \(V(x) .\) State the restrictions. b) Fully factor \(V(x) .\) State the relationship between the factored form of the equation and the graph.

The volume of water in a rectangular fish tank can be modelled by the polynomial \(V(x)=x^{3}+14 x^{2}+63 x+90 .\) If the depth of the tank is given by the polynomial \(x+6,\) what polynomials represent the possible length and width of the fish tank?

a) Given the function \(y=x^{3},\) list the parameters of the transformed polynomial function \(y=\left(\frac{1}{2}(x-2)\right)^{3}-3\) b) Describe how each parameter in part a) transforms the graph of the function \(y=x^{3}\) c) Determine the domain and range for the transformed function.

The product, \(P(n),\) of two numbers is represented by the expression \(2 n^{2}-4 n+3,\) where \(n\) is a real number. a) If one of the numbers is represented by \(n-3,\) what expression represents the other number? b) What are the two numbers if \(n=1 ?\)

The competition swimming pool at Saanich Commonwealth Place is in the shape of a rectangular prism and has a volume of \(2100 \mathrm{m}^{3} .\) The dimensions of the pool are \(x\) metres deep by \(25 x\) metres long by \(10 x+1\) metres wide. What are the actual dimensions of the pool?

A boardwalk that is \(x\) feet wide is built around a rectangular pond. The pond is 30 ft wide and 40 ft long. The combined surface area of the pond and the boardwalk is \(2000 \mathrm{ft}^{2} .\) What is the width of the boardwalk?

A design team determines that a cost-efficient way of manufacturing cylindrical containers for their products is to have the volume, \(V\) in cubic centimetres, modelled by \(V(x)=9 \pi x^{3}+51 \pi x^{2}+88 \pi x+48 \pi,\) where \(x\) is an integer such that \(2 \leq x \leq 8 .\) The height, \(h,\) in centimetres, of each cylinder is a linear function given by \(h(x)=x+3\) a) Determine the quotient \(\frac{V(x)}{h(x)}\) and interpret this result. b) Use your answer in part a) to express the volume of a container in the form \(\pi r^{2} h\) c) What are the possible dimensions of the containers for the given values of \(x ?\)

Determine the equation with least degree for each polynomial function. Sketch a graph of each. a) a cubic function with zeros -3 (multiplicity 2 ) and 2 and \(y\) -intercept -18 b) a quintic function with zeros -1 (multiplicity 3 ) and 2 (multiplicity 2) and \(y\) -intercept 4 c) a quartic function with a negative leading coefficient, zeros -2 (multiplicity 2) and 3 (multiplicity 2), and a constant term of -6

Consider the polynomial \(f(x)=a x^{4}+b x^{3}+c x^{2}+d x+e,\) where \(a+b+c+d+e=0 .\) Show that this polynomial is divisible by \(x-1\)

The width of a rectangular prism is \(w\) centimetres. The height is \(2 \mathrm{cm}\) less than the width. The length is \(4 \mathrm{cm}\) more than the width. If the magnitude of the volume of the prism is 8 times the measure of the length, what are the dimensions of the prism?