To find the lateral area of the right triangular prism, we need to calculate the area of each of the three rectangular faces and then add them together.

Step 1: Calculate the base of the right triangular prism.
The hypotenuse of the right triangular base is given as 30 cm, and the angle is given as 60 degrees. We can use trigonometry to find the lengths of the other two sides of the triangle.

Let's call the two sides adjacent to the angle "a" and "b". The formula for finding the length of a side in a right triangle is:

a = hypotenuse * cos(angle)
b = hypotenuse * sin(angle)

Using this formula, we can calculate the lengths of sides "a" and "b":

a = 30 cm * cos(60 degrees)
a = 30 cm * 0.5
a = 15 cm

b = 30 cm * sin(60 degrees)
b = 30 cm * √3/2
b = 15√3 cm

So, the base of the right triangular prism is a rectangle with sides measuring 15 cm and 15√3 cm.

Step 2: Calculate the lateral area of each rectangular face.
The lateral area of a rectangle is given by the formula: area = length * width.

For the first rectangular face, the length is 25 cm (the height of the prism) and the width is 15 cm. So, the area of the first face is:

Area1 = 25 cm * 15 cm
Area1 = 375 cm^2

For the second rectangular face, the length is 25 cm (the height of the prism) and the width is 15√3 cm. So, the area of the second face is:

Area2 = 25 cm * 15√3 cm
Area2 = 375√3 cm^2

For the third rectangular face, the length is 15 cm and the width is 15√3 cm. So, the area of the third face is:

Area3 = 15 cm * 15√3 cm
Area3 = 225√3 cm^2

Step 3: Calculate the total lateral area of the prism.
To find the total lateral area, we add up the areas of the three rectangular faces:

Total Lateral Area = Area1 + Area2 + Area3
Total Lateral Area = 375 cm^2 + 375√3 cm^2 + 225√3 cm^2

Now, we can calculate the numerical value of the total lateral area using a calculator:

Total Lateral Area ≈ 178.52 cm^2

Therefore, the lateral area of the right triangular prism is approximately 178.52 cm^2.

Question

To find the lateral area of the right triangular prism, we need to calculate the area of each of the three rectangular faces and then add them together.

Step 1: Calculate the base of the right triangular prism.
The hypotenuse of the right triangular base is given as 30 cm, and the angle is given as 60 degrees. We can use trigonometry to find the lengths of the other two sides of the triangle.

Let's call the two sides adjacent to the angle "a" and "b". The formula for finding the length of a side in a right triangle is:

a = hypotenuse * cos(angle)
b = hypotenuse * sin(angle)

Using this formula, we can calculate the lengths of sides "a" and "b":

a = 30 cm * cos(60 degrees)
a = 30 cm * 0.5
a = 15 cm

b = 30 cm * sin(60 degrees)
b = 30 cm * √3/2
b = 15√3 cm

So, the base of the right triangular prism is a rectangle with sides measuring 15 cm and 15√3 cm.

Step 2: Calculate the lateral area of each rectangular face.
The lateral area of a rectangle is given by the formula: area = length * width.

For the first rectangular face, the length is 25 cm (the height of the prism) and the width is 15 cm. So, the area of the first face is:

Area1 = 25 cm * 15 cm
Area1 = 375 cm^2

For the second rectangular face, the length is 25 cm (the height of the prism) and the width is 15√3 cm. So, the area of the second face is:

Area2 = 25 cm * 15√3 cm
Area2 = 375√3 cm^2

For the third rectangular face, the length is 15 cm and the width is 15√3 cm. So, the area of the third face is:

Area3 = 15 cm * 15√3 cm
Area3 = 225√3 cm^2

Step 3: Calculate the total lateral area of the prism.
To find the total lateral area, we add up the areas of the three rectangular faces:

Total Lateral Area = Area1 + Area2 + Area3
Total Lateral Area = 375 cm^2 + 375√3 cm^2 + 225√3 cm^2

Now, we can calculate the numerical value of the total lateral area using a calculator:

Total Lateral Area ≈ 178.52 cm^2

Therefore, the lateral area of the right triangular prism is approximately 178.52 cm^2.

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