Tutor profile: Maddie A.
If the shadow of a building increases by 5 meters when the angle of elevation of the sun rays decreases from 60 degrees to 50 degrees, what is the height of the building?
Before we move on to solving for the height of the building, we should first understand what the angle of elevation of the sun is. The angle of elevation is the angle that the sun’s rays form when measured from the ground or from a horizontal surface. In our sample question, the shadow of the building on the ground, the building and the sun’s rays form a right angle triangle. To find out the height of the building, we first need to visualize two triangles with the two angles of elevations, 60 degrees and 50 degrees. For the initial condition where the angle of elevation is 60 degrees, the length of the shadow (X) is adjacent to the elevation angle while the building’s height (H) is opposite to it. When the elevation angle decreases to 50 degrees, the new length of the shadow becomes X+5. We can use the tangent function, which is the opposite side divided by the adjacent. Tan 60 degrees = H/X and Tan of 50 degrees = H/ (X+5). We can follow the following steps to solve these two equations. Tan 60 degrees = H/X -> H = X*Tan 60 degrees Tan 50 degrees = H/(X+5) -> H = (X+5)*Tan 50 degrees -> H = X*Tan 50 degrees + 5*Tan 50 degrees X*Tan 60 degrees = X*Tan 50 degrees + 5*Tan 50 degrees (substitution) X*Tan 60 degrees – X*Tan 50 degrees = 5*Tan 50 degrees (collecting like terms) X*(Tan 60 degrees – Tan 50 degrees) = 5*Tan 50 degrees X = 5*Tan 50 degrees / (Tan 60 degrees – Tan 50 degrees) = 11.03 Substituting this value in our first equation will give us the following. H = X*Tan 60 degrees -> H = 11.03*Tan 60 degrees = 19.1 meters
If XY<0, which of the following is bigger? A) (X^2)*(Y^2) B) (X^3)*(Y^3).
A is bigger. For the result of the multiplication value between two numbers to be below zero, one of the numbers should be a negative number and the other number should be a positive number. Thus in this case, either X or Y are negative numbers. For choice A, both X and Y are squared. The square of both positive and negative numbers gives a positive number so we can be sure that the squares of both X and Y are positive numbers. Multiplying a positive number by another positive number will also give us a positive number. For example, if X = 2 and if Y = -2, X^2 = 4 and Y^2 = 4, and 4*4=16. For choice B, Both X and Y are cubed. The cube of a positive number is always positive. In fact, any positive number to the power of an odd number gives a positive value. On the contrary, any negative number to the power of an odd number gives a negative value. Thus if either X or Y are negative numbers, then the multiplication of the cube of X and Y will give a negative number. For example, For example, if X = 2 and if Y = -2, X^3 = 8 and Y^3 = -8, and 8*(-8) = -64.
If the price of coffee at your local coffee shop increases from $3 to $4, by how many percent did the price increase?
To calculate the percentage increase, you first subtract the original value from the new value and then divide it by the absolute value of the original value. You then multiply this final amount by 100%. In the sample question, our original value is $3 and our new value is $4. So we first subtract $3 from $4 which gives us $1. When then divide $1 by the absolute value of $3 which gives us a value of 0.333. the final step is to multiply this value by 100%. This will give us 33.3%. Thus the percentage increase is 33.3%.
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