Question

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Explanation:

Identify given dimensions

We are given a rectangular base with length \(l = 2x + 3\) and width \(w = x + 1\) in metres.

Express and simplify perimeter

Using Order of Operations, the perimeter \(P\) is:

$$ P = 2(l + w) = 2((2x + 3) + (x + 1)) $$

Combine like terms inside:

$$ P = 2(3x + 4) = 6x + 8 $$

Expand and simplify area

The area \(A\) is the product of length and width:

$$ A = l \times w = (2x + 3)(x + 1) $$

Expand using the distributive property:

$$ A = 2x(x) + 2x(1) + 3(x) + 3(1) = 2x^2 + 5x + 3 $$

Solve for x and find dimensions

Set the perimeter expression equal to \(44\):

$$ 6x + 8 = 44 \implies 6x = 36 \implies x = 6 $$

Substitute \(x = 6\) to find the dimensions:

$$ \text{Length} = 2(6) + 3 = 15\text{ m} $$

$$ \text{Width} = (6) + 1 = 7\text{ m} $$

Calculate the volume

The volume \(V\) with a uniform depth of \(1.5\text{ m}\) is:

$$ V = \text{Area} \times \text{depth} = (15 \times 7) \times 1.5 = 105 \times 1.5 = 157.5\text{ m}^3 $$

Answer:

Question (a)

An expression for the perimeter is \(6x + 8\) metres.

Question (b)

An expression for the area is \(2x^2 + 5x + 3\) square metres.

Question (c)

The value of \(x\) is \(6\).

Question (d)

The length is \(15\) metres and the width is \(7\) metres.

Question (e)

The volume of water is \(157.5\) cubic metres.