Answer:
A. Three factors; length, width, and depth = (2x + 5)(x + 3)(x - 1)
B. Length = 2x + 5; x = 4; Dimensions = 13 ft × 7 ft × 3 ft; V = 273 ft³
C. See below
Step-by-step explanation:
A. Synthetic division
2 11 15
1)2 9 4 -15
2 11 15
2 11 15 0
So, (2x³ + 9x² + 4x – 15)/(x – 1) = 2x² + 11x + 15
There should be three factors, corresponding to length, width, and depth of the tank.
The depth of the tank is x - 1.
The other two factors are the length and the width of the tank. They are the factors of the quadratic.
x² + 11x + 15 = (2x+5)(x+3)
So, the linear factors of the equation are (x-1)(2x+5)(x+3).
B. Volume of tank
(i) Identify the length factor.
2x + 5 = 2 × 13 +5 = 31
x + 3 = 13 + 2 = 16
l > w, so l = 2x+5
(ii) Value of x
l = 13, so
2x + 5 = 13 Subtract 5 from each side
2x = 8 Divide each side by 2
x = 4
(iii) Dimensions of tank
l = 13 ft
w = x + 3
= 4 + 3
= 7 ft
d = x – 1
= 4 – 1
= 3 ft
The tank dimensions are 13 ft × 7 ft × 3 ft.
(iv) Volume of tank
V = lwd
= 13 × 7 × 3
= 273 ft³
(v) Check
Insert the value of x into the cubic equation.
V = 2x³ + 9x² + 4x – 15
= 2(4)³ + 9(4)² + 4(4) – 15
= 128 + 144 + 16 – 15
= 273
Substituting into the cubic gives the same volume as substituting into its factors.
(C) Graph
The graph of the function is shown below.
Only values in the first quadrant are important, because the dimensions and the volume must be positive.
The y-intercept at x = 1 corresponds to the factor representing the
depth (d = x – 1).
The green dot represents the volume when x = 4.
