If the length (L) of the prism is 10 inches, and each rectangular face has a width (W) 2�ã3 of inches, determine the total surface area of the prism in square inches. Round your answer to the nearest inch.
the three sides: 3L*W total
the two ends: area=w^2 sqrt3 /4
To determine the total surface area of the prism, we need to find the surface area of each rectangular face and then add them together.
Step 1: Calculate the surface area of one rectangular face.
The formula for the surface area of a rectangle is: S = L * W
Given that the length (L) is 10 inches and the width (W) is 2�ã3 inches, we can calculate the surface area of one rectangular face:
S = 10 inches * 2�ã3 inches
To multiply a mixed number, convert it to an improper fraction:
2�ã3 inches = (2*3 + 1) / 3 inches = 7/3 inches
S = 10 inches * 7/3 inches = (10 * 7) / 3 inches² = 70/3 inches²
Step 2: Calculate the surface area of another rectangular face.
Since all rectangular faces of the prism are identical, the surface area of this face will be the same as the one calculated in step 1:
S = 70/3 inches²
Step 3: Calculate the surface area of the remaining four rectangular faces.
Since there are four identical rectangular faces, we need to multiply the surface area of one face by 4:
S = 4 * (70/3 inches²) = (4 * 70) / 3 inches² = 280/3 inches²
Step 4: Calculate the total surface area by summing up the surface area of all the faces.
To find the total surface area, we need to add the surface area of the two identical rectangular faces (calculated in step 2) to the surface area of the four remaining identical rectangular faces (calculated in step 3):
Total surface area = (70/3 inches²) + (280/3 inches²)
To add fractions, they need to have a common denominator:
Total surface area = (70/3 inches²) + (280/3 inches²) = (70 + 280) / 3 inches² = 350/3 inches²
Step 5: Round the answer to the nearest inch.
To round the surface area to the nearest inch, divide the numerator (350) by the denominator (3) and round to the nearest whole number:
Total surface area = 350/3 inches² = 116.67 inches² (rounded to the nearest inch)
Therefore, the total surface area of the prism is approximately 117 square inches.
To determine the total surface area of the prism, we need to calculate the surface area of each rectangular face and then sum them up.
The prism has 2 identical rectangular faces on the top and bottom, and 3 identical rectangular faces on the sides.
Let's start by calculating the surface area of one rectangular face.
The formula to find the surface area of a rectangle is A = L x W, where A is the area, L is the length, and W is the width.
In this case:
L = 10 inches (given)
W = 2�ã3 inches
To convert the mixed number 2�ã3 into a decimal, we can use the fact that 3/3 is equivalent to 1. Therefore, 2�ã3 is equal to 2 + 1/3, which is equal to 2.33 when rounded to 2 decimal places.
Now we can calculate the surface area of one rectangular face:
A = L x W
A = 10 inches x 2.33 inches
A = 23.3 square inches
Since there are 2 identical rectangular faces on the top and bottom, the total surface area contributed by these faces is 2 x 23.3 = 46.6 square inches.
Next, let's calculate the surface area contributed by the 3 identical rectangular faces on the sides.
Again, the width is 2�ã3 inches or 2.33 inches.
The length, in this case, is equal to the perimeter of the base rectangle, which can be calculated by adding the width to the length twice (since the perimeter of a rectangle is the sum of all four sides).
Perimeter = 2L + 2W
Perimeter = (2 x 10 inches) + (2 x 2.33 inches)
Perimeter = 20 inches + 4.66 inches
Perimeter = 24.66 inches
Now we can calculate the surface area of one side rectangular face:
A = L x W
A = 24.66 inches x 2.33 inches
A = 57.4258 square inches
Since there are 3 identical rectangular faces on the sides, the total surface area contributed by these faces is 3 x 57.4258 = 172.2774 square inches.
Finally, we can find the total surface area of the prism by summing up the surface area of all the faces:
Total Surface Area = Top and Bottom Surface Area + Side Surface Area
Total Surface Area = 46.6 square inches + 172.2774 square inches
Total Surface Area = 218.8774 square inches
Rounding this answer to the nearest inch, the total surface area of the prism is approximately 219 square inches.