If f(x) = x/3 - 5 , find the inverse of f(x).
In order to find the inverse function, let's first write this equation in a nicer way. f(x) is another term for y, so we can write this as y = x/3 - 5. The inverse function is exactly as it sounds, a function that is inverted. Instead of solving for y, we need to instead solve for x. So if y = x/3 - 5 then y + 5 = x/3 3*(y+5) = x simplified, this looks like 3y + 15 = x. This is our new inverse function, but we need to rewrite it in standard terms. Because x is always our dependent variable, we need to switch y out for x. So our inverse function is 3x + 15
An 8 foot metal guy wire is attached to a broken stop sign to secure its position until repairs can be made. Attached to a stake in the ground, the guy wire makes an angle of 51º with the ground. How far from the foot of the stop sign is the stake, to the nearest tenth of a foot?
For this it is best to first draw out a picture. Once we do this, we see that we're dealing with a triangle of sorts. This triangle has an 8ft hypotenuse (metal wire), and this hypotenuse makes a 51 degree angle with the base of the triangle. We are being asked now to find the length of the base of this triangle (distance on the ground). With respect to this angle, we are working with an adjacent side and a hypotenuse. Knowing our trig equations, it is clear that in order to find the missing side we will have to use cosine (adjacent/hypotenuse). cos(51º) = adj / 8 so adj = 8*cos(51º) Now we plug this into our calculator and find that the adjacent side (adj) is equal to about 5.03. Rounded to the nearest tenth, this side is 5.0 ft long.
A pulley can move 6.7 feet of rope per minute. How many feet of rope will the pulley move in 8 minutes?
For this question we first need to break the problem down into steps. Because this is a rate problem (the word 'per' makes this clear), we will need to set up a proportion. The pulley can move 6.7 feet of rope per minute -- we can write this as 6.7 ft / 1 min. Now we are being asked to use this rate to find out how many feet we can pull in 8 minutes. Our unknown here is the number of feet, and we want this number to be proportional to our rate. So our equation will look like 6.7 ft / 1 min = x ft / 8 min. Now, because we have two fractions set equal to each other, we can cross multiply which gives us 8(6.7) = 1(x) or 53.6 ft = x. Therefore the pulley will move 53.6 ft in 8 minutes.