Surface Area of a Cuboid: Using ratios for calculation

To solve this problem, we'll follow these steps:

• Step 1: Express dimensions in terms of a single variable using the given ratios.

• Step 2: Substitute these expressions into the cuboid's surface area formula.

• Step 3: Solve for the variable and in turn calculate the dimensions.

Now, let's work through each step:

Step 1: Express dimensions using the ratios.
- Let $h = 3x$ (height is set from height to width ratio), $w = 5x$ (width), $l = \frac{7}{2}h = \frac{7}{2}(3x) = \frac{21}{2}x$ (length from 2:7 as basis relative to height).

Step 2: Substitute into the surface area formula.
The surface area $S$ is given by:
$S = 2(lw + lh + wh) = 542 \, \text{cm}^2$

Plugging in, we get:
$2\left(\frac{21}{2}x \cdot 5x + \frac{21}{2}x \cdot 3x + 3x \cdot 5x\right) = 542$

Simplifying further:
$2\left(\frac{105}{2}x^2 + \frac{63}{2}x^2 + 15x^2\right) = 542$

Combining terms, we get:
$2 \times \left(\frac{105x^2 + 63x^2 + 30x^2}{2}\right) = 542$

The combined denominator cancels out:
$198x^2 = 542$

Solving for $x^2$:
$x^2 = \frac{542}{198} \approx 2.737$

Solving for $x$:
$x \approx \sqrt{2.737} \approx 1.654$

Step 3: Calculate dimensions using $x$:
- $h = 3x \approx 3 \times 1.654 \approx 4.96 \, \text{cm}$
- $w = 5x \approx 5 \times 1.654 \approx 8.26 \, \text{cm}$
- $l = \frac{21}{2}x \approx \frac{21}{2} \times 1.654 \approx 17.36 \, \text{cm}$

Therefore, the solution to the problem is $\text{Height } = 4.96 \, \text{cm}, \text{ Length } = 17.36 \, \text{cm}, \text{ Width } = 8.26 \, \text{cm}$, which matches choice 4.

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.

To solve this problem, we'll follow these steps:

• Step 1: Solve for the variable $x$ using the given ratio and dimension.
• Step 2: Calculate the dimensions of the cuboid.
• Step 3: Apply the surface area formula for a cuboid.
• Step 4: Carry out the calculations to find the total surface area.

Now, let's work through each step:

Step 1: Solving for $x$

We know that the dimensions of the cuboid are in the ratio $1:2:4$. Assigning the dimensions as $x$, $2x$, and $4x$, and knowing that the middle dimension is 5 cm, we have $2x = 5$. Solving for $x$, we get:

$x = \frac{5}{2} = 2.5 \text{ cm}$

Step 2: Calculate the dimensions

Now, using the value of $x$:

• First dimension: $x = 2.5 \text{ cm}$
• Second dimension: $2x = 5 \text{ cm}$
• Third dimension: $4x = 10 \text{ cm}$

Step 3: Apply the surface area formula

The formula for the surface area of a cuboid is:

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

Substitute the dimensions into the formula:

$A = 2(2.5 \times 5 + 2.5 \times 10 + 5 \times 10)$

Step 4: Perform the calculations

Calculate each term:

$2.5 \times 5 = 12.5$

$2.5 \times 10 = 25$

$5 \times 10 = 50$

Now, calculate the total inside the parenthesis:

$A = 2(12.5 + 25 + 50) = 2(87.5)$

$A = 175 \text{ cm}^2$

Therefore, the surface area of the cuboid is $\mathbf{175 \text{ cm}^2}$.