word problem equations and inequalities

Question

Volume of a Trough

Water is running into a trapezoidal trough. The volume V, in cubic feet of water in the trough is given by V=32d^2+32d, where d is the depth in feet of the water in the trough. Find the depth of the water when the volume is 850 cubic feet. Round to the nearest tenth of a foot.

Answer 1

Let's plug what we know into the equation.

V = 32d^2 + 32d

V = 850

850 = 32d^2 + 32d.

We're soling for d, the depth of the water in feet. Since d appears twice this would be challenging on its own, but we can adjust the formula to look like a quadratic equation:

0 = 32d^2 + 32d - 850

Then solve using the following formula:

(-b +- sqrt(b^2 - 4(ac)))/2a

where a = 32, b = 32, and c = -850.

(-32 +- sqrt(32^2 - 4(32)(-850)))/2*32

(-32 +- sqrt(109,824))/64

Divide into the plus and minus halves:

(-32 + sqrt(109,824))/64 = 4.68

(-32 - sqrt(109,824))/64 = -5.68 <-negative feet doesn't make sense for this problem.

After rounding to the nearest 10th of a foot, we find that depth = 4.7ft.