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disa
2 months ago
7

A cone is cut out of the center of a pyramid with a rectangular base. The rectangular base has side lengths of 15 and 10 units.

The cone has a diameter of 9 and a height of 12. The pyramid and cone have the same height. What is the volume of the shaded portion of the composite figure? Express your answer in terms of π. 519π units3 681π units3 (600π – 81) units3 (600 – 81π) units3

Mathematics
1 answer:
Svet_ta [12.7K]2 months ago
3 0

Answer:

Volume of the shaded area = (600 - 36π) units³

Step-by-step explanation:

Volume of the shaded area = Volume of pyramid - Volume of cone

Volume of pyramid = ⅓*l*w*h

Where,

l = length of the base of the pyramid = 15 units

w = width of the base of the pyramid = 10 units

h = height of pyramid = 12 units

Substituting the values helps find the volume of the pyramid

Volume of pyramid = ⅓*15*10*12 = 5*10*12 = 600 units³

Volume of Cone = ⅓πr²h,

Where,

r = radius = ½ of diameter = ½ of 9 = 3 units

h = height = 12 units

Volume of Cone = ⅓*π*3²*12 = ⅓*π*9*12

= π*3*12 = 36π units³

Volume of shaded area = (600 - 36π) units³

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A high school wants to sell postage stamps with the school logo on them. The fundraising group have purchased 500 sheets of stam
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a) Expense

C(q) = 50+16q\\\\C(500)=50+16(500)=50+8,000=8,050

b) Revenue from sales

S(q)=20q\\\\S(500)=20\cdot 500 = 10,000

c) Values table

\left[\begin{array}{ccc}q&C(q)&S(q)\\0&50&0\\250&4,050&5,000\\500&8,050&10,000\end{array}\right]

d) Included

e) Breakeven point = 12.5 sheets

f) Earnings at 550 sheets = $1,950

In-depth analysis:

a) There is a fixed cost of $50 for the image.

Additionally, there is a variable cost of $16 for each sheet.

The total quantity purchased is 500 sheets.

Thus, the cost function can be established as:

C(q) = 50+16q\\\\C(500)=50+16(500)=50+8,000=8,050

b) Each sheet sells for $20, leading to:

S(q)=20q\\\\S(500)=20\cdot 500 = 10,000

c) Values table

\left[\begin{array}{ccc}q&C(q)&S(q)\\0&50&0\\250&4,050&5,000\\500&8,050&10,000\end{array}\right]

d) Included

e) To avoid losses, the minimum number of sheets that must be sold is determined by the breakeven point (BEP), calculated by equating sales income to costs:

S(q)=C(q)\\\\20q=50+16q\\\\(20-16)q=50\\\\4q=50\\\\q=50/4=12.5

f) Profit is computed as the difference between sales income and the cost:

P(500)=S(500)-C(500)\\\\P(500)=20\cdot 500-(50+16\cdot 500)=10,000-8,050=1,950

3 0
2 months ago
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