b Use the formulas for the total surface area and volume of a cone.
A
aTotal Surface Area: 840in.^2
Volume: 1008in.^3
B
bTotal Surface Area: 282.7 m^2
Volume: 314.2 m^3
Practice makes perfect
a Let's first calculate the total surface area and then the volume.
Total Surface Area
The given solid is a prism. To calculate the total surface area of a prism, we can use the known formula where P is the perimeter of the base, h is the height, and B is the area of the base.
S=Ph+2B
We are given two out of three of base's sides. Let's take a closer look at the base.
Notice that the base is a right triangle, so we can relate the three lengths using the Pythagorean Theorem.
a^2+ b^2 = c^2
In the equation above, a and b are the triangle's legs and c is the hypotenuse. In our case, the triangle has legs of 7 and x inches and the hypotenuse is 25 inches. Let's substitute these values into formula above and solve for x.
Now that we know the lengths of all the base's sides. Let's add them to find the perimeter of the base.
P=7+24+25=56inches
Note, the base is a right triangle, so we can calculate its area using the formula for area of a triangle.
The area of the base is 84 in.^2. Now we have enough information to find the surface area of the prism. Let's substitute P with 56, h with 12, and B with 84 into the formula.
The volume of a prism can be calculated using the following formula.
V=Bh
Earlier we calculated that the area of the base B is 84in.^2. We are also given that the height equals 12''. Let's substitute these values into the formula and solve for V.
b Let's first calculate the total surface area and then the volume.
Total Surface Area
The given solid is a cone. To calculate the total surface area of a cone, we can use the known formula where r is the radius of the base and l is the slant height of the cone.
S=π rl+π r^2
From the diagram, we know that the diameter of the cone's base is 10m. By dividing by 2, we get the radius of the base.
r=10/2=5m
To find the slant height, we can use the Pythagorean Theorem. When doing this, the slant height l is the hypotenuse. The height and apothem, which is half of the side length, of the cone are the legs. Let's use these given values to solve for l.
The total surface area of the cone is approximately 282.7m^2.
Volume
To calculate the volume of a cone, we can use the following formula.
V= 13π r^2 h
Here r is the radius and h is the height of the cone. We are given that the height of the cone is 12m and, earlier, we found that the radius of the base is 5m. Let's substitute these values into the above formula and calculate the volume.
