Bob has 500 feet of fencing that he wants to use to fence in his backyard. What should be the dimensions of the fencing in order to have the largest area possible?
First, draw a picture of the backyard and label the sides as X and Y. We are solving for X and Y. You know that the area of the rectangle can be found by solving the formula of A=X*Y. You are trying to maximize A=X*Y. There is a constraint, however, and that is that the perimeter must equal 500 feet. therefore, the constraint can be represented as 500=2X+2Y. Now, the constraint has 2 distinct variables. We can solve the equation so that X=250-Y; this way we can plug this into the Area formula, where we will then have A=(250 - Y)(Y), which can be simplified to A=250Y - (Y^2). Now, we are able to take the derivative of such formula, which would be A'=250 - 2Y. The following step is to equal the equation to 0 and solve for Y. After doing so, we get that Y=125. Plugging this into the Area formula gives us an area of 31250 feet squared. So, according to the method of absolute extrema, this must be the largest possible area since the endpoints equal 0. Now, all that is left to do is solve for the X variable, which can be done by plugging in 125 into the formula of X=250 - Y. We get that X is also 125, therefor the dimensions of the backyard mist be 125 ft x 125 ft.
Jason, standing on the ground, throws a ball vertically in the air with an initial velocity of 50 m/s. After time, T seconds, its height above the ground is given as S(T)=50T-(15)T^2. Find its instantaneous velocity after T seconds. Then find its acceleration at T seconds.
To solve this you must take the first derivative of the given equation. S(T)'=50-30T. This is then the formula for the instantaneous velocity after T seconds. To find the acceleration of the ball after T seconds, you must take the derivative of the second derivative; the second derivative. This would be S(T)''= -30. Which means that regardless of T seconds, the acceleration will always be -30.
The Smith family created a square patio with an area of 185 square meters. Between what whole numbers is the length of one side?
To solve this, you must know that 13 squared is 169 and 14 squared is 196. Since 185 is between 169 and 196, its square root must also be between the square root of those two numbers. Therefore, the length of one side is between 13 and 14 meters.