A circle of diameter 6 inches contains a triangle with a vertex A at the center, a vertex B located on the circumference of the circle at an angle of 50 degrees North of West from the center, and a vertex C located on the circumference of the circle at an angle of 10 degrees North of East from the center. Find the arc length s of the arc formed by the circle in between vertex B and vertex C.
First, we need the angle of vertex A, as it determines the size of the arc. Angle A = (90 - 50) degrees + (90 - 10) degrees = 120 degrees. We take the complements of the given angles since the angles are given with respect to the horizontal, and we want the angles with respect to the vertical. Since there are 360 degrees in a circle, the arc length is 1/3 of the circumference of the circle since 120/360 = 1/3. The circumference of the circle is 2*pi*(6in / 2) = 6*pi inches. s = (1/3)(6*pi inches) = 2*pi inches which is roughly equivalent to 6.28 inches.
Pump A can fill a pool in 4 hours. Pump B can fill the same pool in twice the time. How long will it take the pumps to fill the pool if they work together?
If Pump A can fill a pool in 4 hours, this means it can fill the pool a quarter of the way in an hour. If pump B is twice as fast as pump A, pump B will fill the pool halfway in an hour. If the two pumps are working together, we can add their rates together, to get .25 + .5 = .75. This means that the pumps working together can fill three quarters of the pool in an hour. We can use the equation 1 job = .75 job / hour * time to describe the system. Solving for time, we find that it would take the pumps 1.33 hours if they worked together.
A line in the x-y plane goes through the point (-1 ,-2) has a slope of 3. Find the x and y intercepts of the line.
To find the intercepts, we must first find an equation for the line. Since we have a point and a slope, we can immediately find the point slope form equation for the line, y - y1 = m(x - x1). Substituting in our values, the equation is y + 2 = 3(x + 1). To solve for the x intercept, set y equal to 0. After doing this and solving for x, we get x = -1/3. To solve for the y intercept, set x equal to 0. After doing this and solving for y, we get y = 1. x intercept: (-1/3, 0) y intercept: (0, 1)