Sandra used a triangular prism 10 cm long to refract light into the spectrum. Each triangular end had an area of 3.5 cm squared and sides 3 cm, 3 cm and 2.6 cm. What was the surface of the prism?
To find the surface area of the prism, we need to find the area of each face and add them together.
First, let's find the area of one triangular end of the prism. We can use Heron's formula to calculate the area of a triangle given its side lengths. Heron's formula states that the area (A) of a triangle with side lengths a, b, and c is given by:
A = √(s(s-a)(s-b)(s-c))
where s is the semi-perimeter of the triangle, calculated as:
s = (a + b + c) / 2
In this case, the side lengths of the triangle are 3 cm, 3 cm, and 2.6 cm. Let's calculate the semi-perimeter and then use it to find the area of one triangular end.
s = (3 + 3 + 2.6) / 2
s = 8.6 / 2
s = 4.3 cm
Now, let's calculate the area of one triangular end using Heron's formula:
A = √(4.3(4.3-3)(4.3-3)(4.3-2.6))
A = √(4.3(1.3)(1.3)(1.7))
A = √(11.0991)
A ≈ 3.33 cm² (rounded to two decimal places)
Since there are two triangular ends on the prism, we need to multiply this area by 2 to account for both ends:
2 * 3.33 cm² = 6.66 cm²
Now, let's find the area of the rectangular lateral surface of the prism. The lateral surface area of a rectangular prism can be calculated by multiplying the perimeter of the rectangular base by the height. In this case, the perimeter of the rectangular base is the sum of the two sides (3 cm + 3 cm) and the base (10 cm):
Perimeter = 3 cm + 3 cm + 10 cm
Perimeter = 16 cm
The height of the prism is the same as the length of the prism, which is given as 10 cm.
Now, let's calculate the lateral surface area:
Lateral surface area = Perimeter * Height
Lateral surface area = 16 cm * 10 cm
Lateral surface area = 160 cm²
Finally, let's find the total surface area of the prism by adding the areas of the two triangular ends and the lateral surface area:
Total surface area = 6.66 cm² + 160 cm²
Total surface area = 166.66 cm²
Therefore, the surface area of the prism is approximately 166.66 cm².