The lateral area of a prism is 144cm^2 (lateral area is surface area without the bases). If the height is double the width, and the length is triple the width, what is the surface area?

I got the answer wrong; its supposed to be 196. Can anyone explain how to do the problem?

Write the most precise name for each space figure that has the given properties.



three lateral faces that are rectangles

a.

pyramid

c.

triangular prism

b.

cone

d.

hexagonal prism

To solve this problem, we need to break it down into smaller steps. Here's how you can find the surface area of the given prism.

Step 1: Understand the problem and the given information.
We are dealing with a prism, and we are given the lateral area. The lateral area is the surface area without the bases. Additionally, we are provided with a relationship between the dimensions: the height is double the width, and the length is triple the width.

Step 2: Define the variables.
Let's use the variable "w" to represent the width.

Step 3: Formulate equations using the given information.
Since the height is double the width, the formula for the height can be written as:
height = 2w

Similarly, the length is triple the width, so the formula for the length becomes:
length = 3w

Step 4: Determine the formula for the lateral area.
The formula for the lateral area of a prism is given by the equation:
lateral area = perimeter of the base × height

In this case, the perimeter of the base is the sum of all the sides forming the base. A prism has two congruent bases, so the perimeter is doubled. Since the prism is rectangular, the base perimeter can be calculated as:
perimeter = 2 × (length + width)

Therefore, the formula for the lateral area in terms of width (w) can be derived as:
lateral area = 2 × (3w + w) × 2w
lateral area = 2 × 4w^2
lateral area = 8w^2

Step 5: Solve for the width using the lateral area equation.
Given that the lateral area is 144cm^2, we can substitute this value into the equation and solve for the width (w):
144 = 8w^2
Divide both sides by 8:
18 = w^2
Take the square root of both sides:
w = ±√18

Since width cannot be negative, we take the positive square root:
w = √18

Step 6: Calculate the dimensions and the surface area.
Now that we have the value for the width, we can calculate the height and length using the relationships mentioned earlier:

height = 2w
height = 2√18

length = 3w
length = 3√18

To find the total surface area, we need to find the sum of the lateral area and the areas of the two bases. Since we only know the dimensions of the prism in terms of width, we can calculate the base areas using these dimensions:

base area = length × width
base area = 3√18 × √18
base area = 3 × 18
base area = 54

Now, the surface area can be calculated as:
surface area = lateral area + 2 × base area
surface area = 144 + 2 × 54
surface area = 144 + 108
surface area = 252

Therefore, the correct surface area is not 196 but 252 cm^2.