# A strip of sheet metal 30 cm wide is to be made into a trough by turning strips up vertically along two sides. How many centimeters should be turned up at east side to obtain the greatest carrying capacity?

Mar 14, 2016

$7.5$cm

#### Explanation:

Let the 'turn up' be $t$ $\text{cm}$.

Then twice the cross sectional area in ${\text{cm}}^{2}$ is:

$2 t \cdot \left(30 - 2 t\right) = 60 t - 4 {t}^{2} = {15}^{2} - \left(4 {t}^{2} - 60 t + {15}^{2}\right) = 225 - {\left(2 t - 15\right)}^{2}$

Now ${\left(2 t - 15\right)}^{2} \ge 0$ for any Real value of $t$. Its minimum possible value $0$ is obtained when $\left(2 t - 15\right) = 0$. That is when $t = 7.5$.

This minimum value of ${\left(2 t - 15\right)}^{2}$ results in the maximum value of $225 - {\left(2 t - 15\right)}^{2}$.

Here's a graph of $\frac{1}{10}$ cross sectional area against 'turn up':
graph{(225-(2x-15)^2)/20 [-14.83, 25.17, -5.92, 14.08]}