Q) Swimming Pool : The volume of water in a rectangular, in-ground, swimming pool is given by V (x)= x³ + 11 x² + 24x, where V (x) is the volume in cubic feet when the water is x ft high.
(i) Find the dimension of base of pool.
(ii) Use the remainder theorem to find the volume when x = 3 ft.
(iii) If the volume is 100 ft 3 of water, what is the height x?
(iv) If the maximum capacity of the pool is 520 ft 3 what is the maximum depth?

Ans:

STEP BY STEP SOLUTION

1. Dimensions of Pool’s base:

It is given that the volume of the water V(x) = x³ + 11 x² + 24x

∴ V(x) = x (x² + 11 x + 24)

= x (x² + 8 x+ 3 x + 24)     (by mid term splitting)

= x [x (x + 8) + 3 ( x + 8)]

= x (x + 8) (x + 3)

Now, we know that a cuboid’s volume is given by L x B x H, where L is length, B is breadth & h is height

It is given that in V(x), x is the height of the water in the pool

Therefore, in expression V(x) = x (x + 8) (x + 3); x is height, (x + 8) will be length and (x + 3) will be breadth

at zero water inside the pool, x = 0; and hence length of pool’s base is (0 + 8) = 8 ft

and breadth of the pool’s base is (0 + 3) = 3 ft

Therefore, the pool’s base dimensions are: 8 ft and 3 ft.

(ii) Volume at x = 3, by remainder theorem:

In this question, we need to find the value of V(3)             [at x = 3, V(x) = V(3)]

The remainder theorem states that when a polynomial p(x) is divided by a linear polynomial (x – a), then the remainder is equal to p(a). So if, our expression V(x) is divided by (x – 3), then the remainder will be equal to V(3)

∴ at x = 3, V(3) = 3³ + 11 (3)² + 24 (3)

= 27 + 99 + 72 = 198

Therefore, at x = 3, the volume of the water is 198 ft³

(iii) height at 100 ft³ volume:

we have, V(x) = x³ + 11 x² + 24 x

Given that V(x) = 100

∴ x³ + 11 x² + 24 x = 100

∴ x (x + 8) ( x + 3) = 100

Let’s check by hit & trial:

at x = 1, LHS = 1 x 9 x 4 ≠ 100

at x = 2, LHS = 2 x (2 + 8) ( 2 + 3)

= 2 x 10 x 5 = 100  = RHS

Hence, at x = 2, our equation is balanced.

Therefore, at 2 ft height, the volume of water will be 100 ft ³ 

(iv) Maximum depth at maximum capacity:

Here, at maximum capacity, the pool will be completely filled, and water will be at the highest level; hence, x will be the value of its maximum height.

[Note: Hence, when we will see from the top, the height will be termed as depth. So, don’t get confused by depth, while we were using the term “height” so far.]

Let’s find the value of x at this V9x) = 520 ft ³

∴ x³ + 11 x² + 24 x = 520

∴ x (x + 3) ( x + 8) = 5 x 8 x 13

∴ x = 5

Therefore, at the maximum capacity i.e. 520 ft³ of water in the pool, the maximum depth will be 5 ft.

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