You are walking up a 500 meter high hill. The trail has an incline of 12 degrees. How far will you walk to get to the top?
We know that the sine of an angle can be found by dividing the opposite side length by the hypotenuse side length (given: sin(a)=opposite/hypotenuse). We are given the angle measure and the opposite side length: sin(12)=500/hypotenuse. The length of the hypotenuse is the distance you will walk to the top of the hill, which is what we are solving for. We can rearrange our equation to solve for the hypotenuse: hypotenuse=500/sin(12). By using a calculator, we find the sin(12)=0.21, so 500/(0.21) = 2400 meters.
Suppose a water tank in the shape of a right circular cylinder is thirty feet long and eight feet in diameter. How much sheet metal was used in its construction?
The surface area of a cylinder can be calculated using the equation SA=2*(pi)*(radius)*(height). We know that the radius is four (1/2 of the diameter) and the height is thirty feet. So by plugging these values into the equation, we get SA=2*(pi)*(4)*(30)=240pi square feet. This covers the surface area of the cylinder alone, so now we must find the surface areas of the two circular bases. The area of a circle is given as A=(pi)*(radius)^2. So by plugging in our values we have A=(pi)*(4)^2=16pi. Multiply this amount by two to account for both circular bases: 16pi*2=32pi square feet. Now add the surface area of the cylinder to the surface areas of the two circular bases: 240pi + 32pi = 272pi square feet.
What is the instantaneous rate of change of the line f(x)=2x^3 at x=2?
The instantaneous rate of change is the derivative of the function. The derivative, denoted as f'(x), is 6x^2, so f'(2)=6*(2)^2 or 24.