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Brandon L.

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Trigonometry

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Question:

An equilateral triangle has 3 sides with equal lengths and 3 angles with equal degrees. If the total of the interior angles is 180 degrees, what is the largest angle of each triangle when the equilateral triangle is divided into 3 triangles with equal shape and size?

Brandon L.

Answer:

There are 2 methods in solving this question 1. Know that every triangle regardless of shape has a total interior angle degree of 180 degrees. Therefore to find each angle, an equilateral triangle is 180 degrees divided by 3 = 60 degrees. To divide the equilateral triangle into 3 of identical sizes and shape, draw a line from each angle to the middle of the triangle to form 3 isosceles triangles. This halves each angle into 2 equal angles, resulting in remaining 30 degrees. We know that all triangles have a total interior angle of-of 180 degrees. Therefore, if we subtract 2 angles of 30 degrees from 180, we get the value of the largest angle which is 120. Equilateral Triangle 180/3 = 60 degrees Isosceles Triangle 180 - 2(30) = 120 degrees 2. Know that the total degree of a circle is 360. Imagine fitting a circle into the equilateral triangle and draw one line from each angle to the center of the circle. With this in mind, we divided the equilateral triangle into 3 equal isosceles triangles and will have 3 equally divided sectors. Hence, we divide 360 with 3 to obtain 120 degrees Circle 360 Isosceles Triangle 360/3 = 120 degrees

Basic Math

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Question:

A mother and daughter were having a conversation. There was a time when I was half the age of your father. However, the next year, he was only one and a half times my age. If the mother is 40, how old is the father?

Brandon L.

Answer:

Let the age of mother be X and the age of father be Y At a time X = 1/2 (Y) ---- Equation 1 The following year, X + 1 = X1 Y + 1 = Y1 1.5 (X1) = (Y1) ---- Equation 2 Solve both equations simultaneously by inserting Equation 1 into 2 1.5 [ 1/2(Y) + 1] = Y + 1 3/2 [ 1/2 (Y) + 1] = Y + 1 3/4 (Y) + 3/2 = Y + 1 Move relative unknown (Y) to one side 3/2 - 1 = Y - 3/4 (Y) 1/2 = 1/4 (Y) ----- Inverse 4 from denominator to numerator when moved to the other side (1/2) (4) = Y 2 = Y Putting 2 = Y into Equation 1, X = 1/2 (2) X = 1 The difference in X (1) and Y (2) is 1, meaning the age difference is 1 year apart Hence, when the mother is 40, the father would be 41

Algebra

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Question:

The population of city X changes according to the exponential function X(t) = 2.5 (2)^0.10t (millions) and the population of city Y changes according to the exponential function Y(t) = 1.3 (2)^0.17t (millions) When t = 0 corresponds to the year 2016. a) Which city had a smaller population in the year 2016? b) When will the sizes of the populations of the two cities be equal?

Brandon L.

Answer:

a) X(0) = 2.5 millions , Y(0) = 1.3 millions, city Y had smaller population b) Solve X(t) = 2.5 (2)^0.10t = Y(t) = 1.3 (2)^0.17t to find t Take ln on both sides ln 2.5 (2)^0.10t = ln 1.3 (2)^0.17t ln (2.5) + ln (2)^0.10t = ln (1.3) + ln (2)^0.17t ln (2.5) + 0.10t ln (2) = ln (1.3) + 0.17t ln (2) ln (2.5) - ln (1.3) = 0.17t ln (2) - 0.10t ln (2) [ ln (2.5) - ln (1.3) ] = 0.07t ln (2) [ ln (2.5) - ln (1.3) ] / 0.07 ln 2 = t t = 13.48 When t = 0 (The Year 2016) , hence when t = 13.48 , 2016 + 13.48 = 2029.48 Rounding to nearest year (0.48<0.50), it will be the year 2029 when the population in both cities are equal

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