A person 100 feet from the base of a tree. They observe that the angle between the ground and the top of the tree is 18 degrees. Estimate the height of the tree to the nearest tenth of a foot.
It is helpful in this type of problems to draw a picture. Drawing a picture will help us realize we need to use the Tangent function. Remember, tanget = opposite over hypotenuse in SOHCAHTOA. Using the tangent function gives us: tan(18) = x/100 Solving for x gives us: x = 100 tan(18) x = 32.5 The estimated height of the tree is 32.5 feet.
Two angles are supplementary and one of them is three times as big as the other. What is the size of the larger of the two angles?
We need to remember supplementary angles add up to 180 degrees. We need to write an equation that represents this problem using a single variable. First, we will list the angles and assign them variables. If we use x for the smaller angle, the bigger angle will be 3 times x, represented by 3x. Angle 1: x Angle 2: 3x Since we know the angles are supplementary, we can write the following equation: x + 3x = 180 Solving that equation gives us that x = 45 x + 3x = 180 4x = 180 x = 45 Now we need to answer the question asked in the problem. Since the question is asking for the larger of the two angles, we will plug x = 45 into Angle 2. 3x = 3(45) = 135 degrees. The larger angle is 135 degrees.
What is the solution for the following system of equations? x + y = 8 2x + 3y = 19
There are various methods we could use to solve the above system (graphing, substitution, and elimination). Since the variables line up one on top of the other, the elimination method is the best method. We need try to make one of the variables become opposites (either the x's or the y's). If we wanted to eliminate the x-values first, we would multiply the first equation by -2. If we wanted to eliminate the y-values first, we would multiply the first equation by -3. Let's multiply the first equation by -2 (smaller numbers make it easier to work with). Multiplying the first equation by two gives us the following system: -2x - 2y = -16 2x + 3y = 19 Adding the equations together gives us: y = 3 Remember, the answer to a system of equations is always an ordered pair. This means we need to find the x-value as well. We solve this by plugging the value for y (3) into one of the original equations. Again, let's look for the easiest one. We will plug it into x + y = 8. x + 3 = 8. Solving for x shows that x = 5. This means the solution to this system of equations is (5, 3). We can check this by plugging the ordered pair into the system and making sure it works for both equations. 5 + 3 = 8 (TRUE) 2(5) + 3(3) = 19 (TRUE) This proves our answer works and we did the steps correctly for this system.