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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
<u>2 11 15 </u>
1)2 9 4 -15
<u> 2 11 15 </u>
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) <em>Identify the length factor. </em>
2x + 5 = 2 × 13 +5 = 31
x + 3 = 13 + 2 = 16
l > w, so l = 2x+5
(ii) <em>Value of x </em>
l = 13, so
2x + 5 = 13 Subtract 5 from each side
2x = 8 Divide each side by 2
x = 4
(iii) <em>Dimensions of tank </em>
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) <em>Volume of tank </em>
V = lwd
= 13 × 7 × 3
= 273 ft³
<em>(v) Check </em>
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) <em>Graph </em>
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.
