a Find the hypotenuse of the base using the Pythagorean theorem. Remember to convert the values from inches to feet.
B
b Use the formula for the volume of the prism. Remember to convert the values from inches to feet.
A
a 107.5ft^2
B
b 27.5ft^3
Practice makes perfect
a The sandwich has the shape of a triangular prism. To calculate the surface area of a prism, we can use the determined formula.
S=Ph+2B
Here P is the perimeter of the base, h is the height, and B is the area of the base. First, let's find the perimeter and area of the base.
Perimeter of the Base
From the diagram we can see that the base is a right triangle. Since we know only the lengths of the legs, we can find the hypotenuse using the Pythagorean theorem.
a^2+b^2=c^2
Substituting a with 6.99 and b also with 6.99, we get the equation for c. Let's solve!
Now we know all the triangle's sides, so let's add them to find the perimeter of the base.
P=6.99+6.99+9.9=23.88ft
The perimeter of the base is 23.88 feet.
Area of the Base
To find the area of a right triangle with legs a and b, we can use the following formula.
B=ab/2
Let's substitute a with 6.99 and b with 6.99, and solve for B.
Now, let's find the height of the sandwich when filled. Two slices of bread are each 8 inches thick and the tuna-cucumber filling is 13.5 inches thick. Adding these values, we get the total height of the sandwich.
h=8+13.5+8=29.5in
Notice that the perimeter and area of the base were calculated in feet, while the height is in inches. In order to calculate the surface area we need for all the measures to be in the same units. Let's convert inches into feet.
1foot/12inches
By multiplying the conversion factor by our measurement, we can convert in into feet.
Now, we have enough information to calculate the surface area of the prism.
S=Ph+2B
Let's substitute the perimeter 23.88 for P, the area of the base 24.4 for B and the height of the base 2.46 for h.
The surface area of the sandwich is about 107.5ft^2.
b From the diagram we can see that the filling of the sandwich also has the shape of a triangular prism. In order to calculate the volume of the prism with the height h and the area of the base B, we can use the following formula.
V=BhThe filling has the same base as the sandwich. Earlier we already calculated the area of the sandwich's base. It is 24.4ft^2. The height of the filling is 13.5in. Since we are asked to calculate the volume in cubic feet, we need to convert the height into feet. We can do it the same way as before, by multiplying the conversion factor by 13.5 in.
