A water tank is 12m high.it has a a rectangular base, and the length is 7m more than the width. if the volume is 1728m3, find the dimensions of the tank
12w(w+7) = 1728
12w^2 + 84w - 1728 = 0
w^2 + 7w - 144 = 0
(w-9)(w+16) = 0
The tank is 12x9x16
To find the dimensions of the water tank, we can use the given information about its volume and height. Let's break down the problem into steps:
Step 1: Determine the volume of a rectangular prism.
The volume (V) of a rectangular prism can be calculated by multiplying the length (L) by the width (W) by the height (H). In this case, the volume is given as 1728m³, so we have the equation:
V = L * W * H = 1728m³
Step 2: Express the length in terms of the width.
According to the question, the length (L) of the tank is 7m more than the width (W). We can write this relationship as:
L = W + 7
Step 3: Substitute the length expression into the volume equation.
Substituting the expression for length from Step 2 into the volume equation from Step 1, we get:
V = (W + 7) * W * H = 1728m³
Step 4: Substitute the given height into the volume equation.
The height (H) of the water tank is given as 12m. Substituting this value into the volume equation, we have:
V = (W + 7) * W * 12 = 1728m³
Step 5: Solve the equation for the width.
Now we can solve the equation from Step 4 to find the value of the width (W). First, divide both sides of the equation by 12:
(W + 7) * W = 144
Step 6: Simplify the equation and rewrite it in quadratic form.
Expanding the equation from Step 5, we have:
W^2 + 7W = 144
Step 7: Rearrange the equation.
To rewrite the equation in quadratic form, we need to move all terms to one side of the equation. Subtracting 144 from both sides, we get:
W^2 + 7W - 144 = 0
Step 8: Factor or use the quadratic formula to solve for the width.
The equation from Step 7 can be factored or solved using the quadratic formula. Factoring the quadratic equation, we find that:
(W + 16)(W - 9) = 0
This means that either W + 16 = 0 or W - 9 = 0. Solving both equations, we get:
W + 16 = 0 -> W = -16 (reject because width can't be negative)
W - 9 = 0 -> W = 9
Step 9: Determine the length.
Since the width (W) is 9, and according to the given information, the length (L) is 7m more than the width, we have:
L = W + 7 = 9 + 7 = 16
Step 10: Determine the dimensions of the water tank.
Based on the calculations from Steps 8 and 9, we found that the width (W) is 9m and the length (L) is 16m. Therefore, the dimensions of the water tank are:
Width = 9m
Length = 16m