Solve for Y: Cuboid Surface Area 66x+56 with Dimensions 4×3x×2y

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

• Step 1: Use the formula for the surface area of a cuboid.
• Step 2: Substitute the dimensions and equate to the given surface area expression.
• Step 3: Solve algebraically for the missing dimension $y$.

Now, let's work through each step:

Step 1: Consider a cuboid with dimensions $l = 4$, $w = 3x$, and $h = 2y$.
The formula for the surface area is $SA = 2(lw + lh + wh)$.

Step 2: Substitute the dimensions into the formula:

$SA = 2(4 \times 3x + 4 \times 2y + 3x \times 2y)$

This simplifies to $SA = 2(12x + 8y + 6xy)$.

Further simplifying, we have $SA = 24x + 16y + 12xy$.

According to the problem, this is equal to $66x + 56$. Therefore, set:

$24x + 16y + 12xy = 66x + 56$

Step 3: Solve the equation:

Rearrange the terms:

$24x + 12xy + 16y = 66x + 56$

$12xy + 16y = 42x + 56$

Factor common terms:

$y(12x + 16) = 42x + 56$

Divide throughout by $(12x + 16)$:

$y = \frac{42x + 56}{12x + 16}$

To further simplify, note that both numerator and denominator can be reduced:

Factor out the greatest common divisor:

$y = \frac{14(3x + 4)}{4(3x + 4)}$

Cancel $(3x + 4)$:

$y = \frac{14}{4} = 3.5$

Therefore, the solution to the problem is $y = 3.5$ cm, matching the correct choice.