An object is dropped from the roof of a 1000 foot tall building. a. Construct the position and velocity equations for the object in terms of t, where t is equal to the amount of seconds that have passed since the object was dropped. b. Calculate the average velocity of the object over time interval t=1 and t=3 seconds. c. How many seconds does it take for the object to hit the ground, to the nearest hundreths place?
a. The position function for an object is equal to s(t)= -16t^2+v(0)+s(0), where v(0) is the initial velocity and s(0) is the initial position. Since the object is starting at rest, the initial velocity, v(0) would be equal to 0. The object is being dropped from a 1000 foot tall building, so the initial position, s(0) would be 1000. The position equation would be s(t)=-16t^2+1000. In order to find the velocity equation, one must take the derivative of the position equation. This would make the velocity equation equal to v(t)=-32t. b. The average velocity can be found by determining the slope of the secant line. In order to find this, plug t=1 and t=3 into s(t). This gives you the ordered pair of (1,984) and (3,952). Use the slope formula of (y2-y1)/(x2-x1). This gives you (952-984)/(3-1). Solving the equation, you get that the average velocity is equal to -16 ft/s. c. When the object hits the ground, s(t) is equal to zero. In order to solve this, set the position equation equal to zero and solve for t. -16t^2+1000=0 -16t^2=-1000 (Subtract 1000 from each side) t^2=(-1000/-16) (Isolate variable t^2 but dividing both sides of the equation by -16) t^2=62.5 (In order to simplify the equation for just t, take the square rot of each side in order to cancel out the square that is affecting the t) t=sqrt(62.5) t=7.91s
Identify the property of logarithms used, and simplify the logarithm. log(base 8)4 + log(base 8)6
Using the Product Property, when adding two logarithms with the same base, one can multiply the two arguements together in order to simplify. The answer would be log(base 8)(6*4), which can be furthur simplified as log(base 8)24.
Factor the polynomial: 3x^2 - 6x-9
First, pull out the greatest common factor that all three terms share. For this problem, it would be three. This leaves you with 3(x^2-2x-3). Next, find two numbers that multiply in order to equal -3 and add to equal -2. These two numbers would be -3 and 1. In order to complete factoring, place an x in each set of parentheses followed by the -3 and 1 respectively. The completed factorization should look like 3(x-3)(x+1).