Simplify this, [(Tan(x) * Cos(x))/(Sin(x))] - Cos^2 (x)
So, Tan(x) = Sin(x)/Cos(x) (Sin(x)/Cos(x) * Cos(x)) = Sin(x) Sin(x) / Sin(x) = 1 And using the Pythagorean identity of Sin^2 (x) + Cos^2 (x) = 1 we find that the remaining 1 - Cos^2 (x) is equal to a final simplified answer of Sin^2 (x)
If a 13 foot ladder whose base is sliding away from a wall at a rate of 3 feet per second and the top of the ladder is 12 feet off the ground, at what rate is the ladder sliding down the wall?
First, it is known that with the Pythagorean Theorem here, a^2 + b^2 = c^2. Then, we plug in what we do know to the equation: a^2 + b^2 = 13^2 because the distance staying constant here is that of the length of the ladder, the hypotenuse. Then we take the derivative of that equation which makes it 2a * dA/dt + 2b * dB/dt = 0. Then plug in 12 for a since that is the distance from the ground and use the Pythagorean Theorem to find that b = 25^2 = 5 and also plug in the rate of the ladder sliding away from the wall 2 * 12 * dA/dt + (2 * 5 * 3) = 0, then simplifying 24 * dA/dt + 30 = 0 and solving for dA/dt, dA/dt = -30/24 = -15/12. This number means that the ladder is sliding down the wall at 15/12 feet per second at the moment that the base of the ladder is 5 feet away from the wall.
If the number a is multiplied by 5 and summed with 4 to equal 39, what number is a?
5a + 4 = 39 5a = 35 a = 7