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Geometry

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Question:

Show that Pi > 3.

Yun Q.

Answer:

Consider a semicircle with radius r from 0 to 180 degrees. The radian is the length of the part of the circumference of the circle that has the same value as r. Furthermore, Pi x radian or ( Pi x r) is the length of the circumference of the semicircle. Now, construct 3 identical equilateral triangles each with side r inside the semicircle. All the bases of the triangles face the semicircle. The sum of the length of the bases is 3 x r. Since the radian is curved, it is therefore subtended at a smaller angle than the 60 degrees associated with the equilateral triangle. As a result, the circumference of the semicircle, Pi x r, is greater than 3 x r associated with the sum of the bases of the 3 equilateral triangles. Therefore, Pi >3.

Pre-Algebra

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Question:

Combine like terms: 2xy - 3x - 6 + 5xy -7y + 8x - 4y - 3y - 9xy + 4

Yun Q.

Answer:

First, rearrange and place like terms together. 2xy + 5xy - 9xy - 3x + 8x - 6 + 4 - 7y - 4y - 3y. Count the terms to verify the numbers are the same as the original. Next, combine like terms by adding the numbers associated with the like terms. For example, since for the xy terms, 2 + 5 - 9 = -2, the combined xy terms becomes -2xy. Performing similar additions results in the final answer of -2xy + 5x - 2 - 14y

Algebra

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Question:

Bill takes 3 hours to mow a lawn area A. Mike takes 4.35 hours to mow a lawn of area 2.5A. How long does it take for both of them together to mow a lawn of area 3.5A? (Assume each person mow with constant speed)

Yun Q.

Answer:

In one hour Bill can mow A/3 area and Mike can mow a 2.5A/4.35 area of lawn. Therefore, in one hour both together can mow a total of ( 1/3 + 2.5/4.35 )A area of lawn. What time, t, would be needed for a total of 3.5A of the area to be mowed? (time in hours) x (rate of work being done per hour) = total work is done ( t hr) x ( 1/3 + 2.5/4.35 )A /hr = 3.5A t = 3.5A / ( 1/3 + 2.5/4.35 )A = 3.85 hr.

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