Surface Area of a Cuboid: Express with Variable l Given 3:5 Height-Width Ratio

To solve this problem, we'll express the dimensions of the cuboid in terms of the length $l$.

• The given ratio of height to width is 3:5, which means $h = \frac{3}{5}w$.
• We are told that the length $l$ is greater than 3 times the height. Therefore, $l = 3h + k$ for some $k > 0$.

For simplicity, if we assume $l = 3h$, then substituting $h = \frac{3}{5}w$ gives $l = 3 \times \frac{3}{5}w = \frac{9}{5}w$.

To express $w$ in terms of $l$, we solve $w = \frac{5}{9}l$.

Now, express the height $h$ in terms of $l$ using the relationship $h = \frac{3}{5} \times \frac{5}{9}l = \frac{1}{3}l$.

These substitutions allow us to express all dimensions in terms of $l$:
Width, $w = \frac{5}{9}l$
Height, $h = \frac{1}{3}l$

The surface area $A$ of the cuboid is given by:

$A = 2(lw + lh + wh)$

Substitute the expressions for $w$ and $h$:

$A = 2\left(l \times \frac{5}{9}l + l \times \frac{1}{3}l + \frac{5}{9}l \times \frac{1}{3}l\right)$

$A = 2\left(\frac{5}{9}l^2 + \frac{1}{3}l^2 + \frac{5}{27}l^2\right)$

Simplify the expression inside the parentheses:

$A = 2\left(\frac{15}{27}l^2 + \frac{9}{27}l^2 + \frac{5}{27}l^2\right)$

$A = 2\left(\frac{29}{27}l^2\right)$

$A = \frac{58}{27}l^2$

Therefore, the expression for the surface area in terms of $l$ is $\frac{58}{27}l^2$. This matches the given answer.