The figure is made up of a cylinder, a cone, and a half sphere. The radius of the half sphere is 3 inches. What is the volume of
the composite figure?
1 answer:
Answer:
The volume of the composite figure is 84π ≅ 263.89 inches³
Step-by-step explanation
- The composite figure consists of :
# Half sphere with radius 3 inches
# Cylinder with radius 3 inches and height 6 inches
# Cone with radius 3 inches and height 4 inches
- The volume of the sphere is 4/3 π r³
∴ The volume of the half sphere = 1/2 × 4/3 π r³ = 2/3 π r³
- The volume of the cylinder is π r² h
- The volume of the cone is 1/3 π r² h
* Now lets solve the problem
- The volume of the half sphere
∵ The radius of the half sphere = 3 inches
∵ The volume of it = 2/3 π r³
∴ The volume = 2/3 × π × (3)³ = 18π inches³
- The volume of the cylinder
∵ The radius of the cylinder = 3 inches
∵ The height of the cylinder = 6 inches
∵ The volume of it = π r² h
∴ Its volume = π × (3)² × 6 = 54π inches³
- The volume of the cone
∵ The radius of the cone = 3 inches
∵ The height of the cone = 4 inches
∵ The volume of it = 1/3 π r² h
∴ Its volume = 1/3 π × (3)² × 4 = 12π inches³
- Add all the volumes to find the volume of the composite figure
∴ The volume = 18π + 54π + 12π = 84π = 263.89 inches³
* The volume of the composite figure is 84π ≅ 263.89 inches³
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y = -k(x-1)
Then a rectangle with x-dimension w will have an area that is the product of this width and the height y = -k((w/2)-1).
area = w(-k(w/2 -1)) = (-k/2)w² +kw
The derivative of area with respect to w will be zero where the area is a maximum:
d(area)/dw = 0 = -kw +k
w = 1 . . . . . . add kw and divide by k
Using the formula for area, we find
area = 1(-k(1/2-1)) = k/2
This value is 1/2 the total area of the triangle we started with.
If the side lengths of your triangle are 8, √80, and √80, the height will be given by the Pythagorean theorem as
h = √((√80)² - (1/2·8)²) = √(80-16) = 8
Your triangle's area will be
triangle area = (1/2)(8)(8) = 32
The maximum area of the inscribed rectangle will be half this value,
16 square units
We know that:
Then:
Therefore:
Simplifying the above result we get:
Answer: