A garden has the shape of an isosceles right triangle. The length of the hypotenuse is 24 feet. What is the area of the garden?
First, we need to draw the triangle. Draw a right triangle with equal legs. The legs are the two sides that are not opposite the right angle. An isosceles triangle means that two sides are congruent, so in a right triangle, the two legs are congruent. In this problem, the hypotenuse (the side opposite the right angle) is given. After drawing the triangle, label the hypotenuse, 24 feet. The two legs can be labeled x because we want to find them so we can find the area. Once we find x, we can use the basic formula for the area of a triangle, A = 1/2bh. In this case, b and h are the same, and we can substitute x in for both of them, or A = 1/2(x)^2 In a right triangle, we know that (a)^2 + (b)^2 = (c)^2 (where a and b are the bases and c is the hypotenuse). We can substitute x in for a and b. Therefore (x)^2 + (x)^2 = (24)^2 If we combine like terms, we have 2(x)^2 = 576. 2(x)^2=576 Then (x)^2=288 if we divide both sides by 2. Since our formula for area is ½((x)^2), then we can substitute in 288 for (x)^2 and get 144 as the area of our triangle. Therefore, the area of the garden is 144 square feet.
Solve the equation: 7x - 3x = 28
When solving equations, we should first simplify both sides of the equation. 7x - 3x = 28 We can subtract two variables. 7x - 3x = 4x, so we know that: 4x = 28 We can then simplify by dividing both sides of the equation by 4. We can do this by using the division property of equality, which states that dividing each side of the equation by a nonzero number produces an equivalent equation. This is also called “balancing the equation”. 4x/4 = 28/4 x=7 (The 4’s cancel out because of 4/4 = 1 Therefore x = 28/4 which is 7, so x=7)
The Rusty Metal Roller Coaster at the fair charges $2.50 for children and $4 for adults. Before the safety inspector shuts down the ride, 260 people ride it and $815 is collected. How many adults rode the coaster?
The first thing you have to do is identify your variables Let c = the number of children and a = the number of adults You know that 200 people ride the roller coaster total. That is the number of children and adults put together. So, c + a = 200 Also, you are given that $815 is collected total. You would get this amount by multiplying the price of a child’s ticket, $2.50 times the number of children, then adding that to the price of an adult ticket, $4, times the number of adults. Therefore, 2.50c + 4a = 815 Now, you have a system of equations, A systems of equations just means a set of equations that are working together. If we can write two equations with two variables, we can put them together to figure out the values for each of the variables. We have: c + a = 200 2.50c + 4a = 815 The best way to solve this equation is a process called substitution. This means we must isolate one of the variables. c + a = 200 - a = 200 - a ____________ c = 200 - a We subtract a from both sides (balancing the equation), and get c = 200 - a We substitute that into the other equation 2.50 (200 - a) + 4a = 815 We distribute and simplify the equation: 2.50*200 - 2.50a + 4a = 815 500 - 2.5a + 4a = 815 500 + 1.5a = 815 Then, we want to isolate the variable. 500 + 1.5 a = 815 -500 = -500 _______________ 1.5a = 315 /1.5 = /1.5 (divide both sides by 1.5) a = 315/1.5 a = 210 Since the equation asks how many adults rode the coaster, we are finished. 210 adults rode the coaster. If we wanted to find out the number of children that rode the equation, we could substitute in 210 for our a variable in either of our equations.