Here is her answer.
V = 2034.72 in.^3 *
We know that the answer is incorrect. We want to find the correct volume of the figure. To do so, note that the composite solid consists of a cylinder and a cone stacked on top of one another.
Then the inner cylinder was hollowed out.
This means that the volume of the figure is the sum of the volume of the outer cylinder V_1 and the volume of the cone V_2 minus the volume of the inner cylinder V_3.
V = V_1+V_2-V_3
Let's find the volumes one by one.
Volume of the Outer Cylinder
We will start with the larger cylinder. We want to find its volume V_1.
To do so, we can use the formula for the volume of a cylinder.
V_1=Ï€ r^2 h
From the diagram we can see that the height h of the cylinder is 15 inches. Also, the diameter of the base is 12 inches. This means that the radius r of the base is 12÷ 2=6 inches. Let's substitute these values into the formula and calculate the volume.
The volume of the outer cylinder is about 1696 cubic inches.
Volume of the Cone
Now let's find the volume of the cone V_2.
Let's recall the formula for the volume of a cone.
V_2= 13Ï€ r^2 h
From the diagram we can see that the height of the cone h is 9 inches. We earlier found the radius of the base of the cylinder to be 6 inches. This means that the radius of the base of the cone r is also 6 inches. Let's substitute the values and then find the volume.
Now we want to find the volume of the hollow cylinder V_3.
From the diagram we know both the height of the cylinder h and the radius of the base r.
h &= 15 in.
r &= 3 in.
Let's substitute these values into the formula for the volume of a cylinder and evaluate.
The volume of the inner cylinder is about 424 cubic inches.
Volume of the Figure
Earlier we noted that the volume of the figure is the sum of the volume of the outer cylinder V_1 and the volume of the cone V_2 decreased by the volume of the inner cylinder V_3.
V = V_1+ V_2- V_3
Let's substitute the volumes for the values that we found. Then we will simplify the right-hand side.
V = 1696+ 339- 424 = 1611
The volume of the figure is about 1611 cubic inches.
We now have two answers to the problem. The correct volume is the one we found. The incorrect volume is the one proposed by the student.
V &≈ 1611 in.^3 ✓
V &= 2034.72 in.^3 *
We want to find the mistake that the student likely made. Let's calculate the difference between the volumes.
2034.72- 1611 &= 423.72
&≈ 424
The difference between the volumes is about 424 cubic inches. This is exactly the volume of the inner cylinder we found in Part A. This means that the student most likely forgot to subtract the volume of the inner cylinder from the volume of the figure.
