# Thread: Polynomial Equations.

1. ## Polynomial Equations.

1.A case is packed with 12 cylindrical cans, each having a diameter of 15 cm and height of 24cm. Calculate the volume of the empty space in the case to the nearest whole number.

2. Sketch the graphs of f(x)= -(x-1)(x+2)(x-4) and g(x)= -(x-1)(x+2)(x+2) and compare their shapes and the number of x-intercepts.

2. ## Part 1 first

Assuming the cans reach from the bottom to the top of the box and they are in a single row then the volume of the box is (12 x15) x 15 x 24 = 64800 cubic cm. The volume of each tin is pi x r squared x h = 4241.15 (2 decimal places) which given there are 12 cans makes a total volume of 50893.80 cubic centimetres. This means that the volume of empty space is 13906.19901 cubic cm. This does assume the cans are packed in a single row in the box and reach the edge of the box in all 3 dimensions, if this is not the case we need to know how they are stacked to give the solution.

3. ## Part 2 now

The - in front of each gives them as negative cubic curves, both appearing from the top left (quadrant 2) and having 2 turning points before disappearing down into the bottom right. The first curve has three distinct solutions at x = -2, 1, and 4, with a y intercept gained by mulitply the numbers -1, 2, and -4 together to give 8, then mulitplying by the - in front to give -8. The second curve has crosses the x axis once at 1 and has a repeated root at x = -2, which means the x axis is a tangent to the curve at that point. This curve crosses the y-axis at +4. Hope this gives you enough information to sketch the curves.