finding two sides of a cuboid while knowing the volume

Question

The volume of a cuboid is $150\,\mathrm m^3$, one side is $2,5\,\mathrm m$, one side is completely unknown ($x$) and the other one is $4$ meters longer ($x+4$). I guessed the sides one being $6$ and the other $10$ but I need to figure it out with a formula.

Answer 1

If you use the volume formula for this cuboid you get $150=2.5x(x+4)$, and you can divide by $2.5$ to get $x(x+4)=60$, which rearranges to a quadratic equation.

Answer 2

The formula for volume is $$V = lwh$$ where $l$ is the length, $w$ is the width, and $h$ is the height.

You are given: $$V = 150$$ $$w = 2.5$$ $$l = x$$ $$h = x + 4.$$

Plugging the values into the formula for volume of a cuboid, you get $$150 = 2.5x(x+4)$$ $$300 = 5x^2 + 20x$$ $$60 = x^2 + 4x$$ $$x^2 + 4x - 60 = 0.$$

Now, this last equation is a quadratic in $x$, which can be solved by factoring: $$(x + 10)(x - 6) = 0.$$ $$x + 10 = 0 \text{ or } x - 6 = 0$$ $$x = -10 \text{ or } x = 6.$$

The root $x = -10$ is extraneous. Therefore, $x = 6$ and $x + 4 = 10$.