City urbanization City planners model the size of their city u...

Question

City urbanization City planners model the size of their city using the function A(t) = - 1/50t² + 2t +20, for 0 ≤ t ≤ 50, where A is measured in square miles and t is the number of years after 2010.
b. How fast will the city be growing when it reaches a size of 38 mi²?

b. How fast will the city be growing when it reaches a size of 38 mi²?

Accepted Answer

Welcome back, everyone. In this problem, we want to determine the rate of change of the reserves area when it reaches 50 square kilometers if the area of a wildlife reserve is given by the function A of T equals -160 of T2 + 4 T + 30 for T between 0 and 60, where A is measured in square kilometers and T is the number of years after 2015. A says it's 6.66 square kilometers per year. B. 7.66 square kilometers per year. C. 4.83 square kilometers per year, and D 3.83 square kilometers per year. Now to determine the rate of change of the reserves area when it reaches 50 square kilometers, we'll need to know the time or the value oft when the area with respect to T is 50 square kilometers, and we'll also need to know the general rate of change, that is the rate of change of a with respect to T. So first, let's solve 4 TU and ART equals 50 square kilometers because we'll need that value, OK? Now, when A of T, is equal to 50 square kilometers, OK, then that means 50 will be equal to -160 of T2 + 40 + 30. Now, we can multiply through by 60 to clear the fraction, OK? And when we multiply our entire equation by 60, then it becomes 3000 equals negative T2, OK, plus 240 T. Plus 1800. Let's rearrange this to the standard form of a quadratic equation. So that's gonna be T squared minus 240T plus 3000 minus 1800 or 1200, and that equals 0. Now we can use the quadratic formula to solve for T because here since A the coefficient of T squared is 1, b the coefficient of T is negative 240, and C or constant is 1200. Then by the quadratic formula, that means, let me keep that in the same color here. T is going to be equal to negative B plus R minus the square root of B2 minus 4AC all divided by 2A substituting our values, that's going to be the negative value of -240 plus R minus the square root. Of -240 squad. minus 4 multiplied by 1, multiplied by 1200, all divided by 2 multiplied by 1, OK? Now if we go ahead and simplify here, then we're going to get T to be equal to 240 plus or minus the square root of 52,800 divided by 2. Which means then that T is going to be equal to 240 plus the square root of 52,800 divided by 2 or 240 minus the square root of 52,800 all divided by 2. When we're adding them, T is going to be equal to 234.89. 243, sorry, sorry, 234, my apologies. 234.89 years. Or when we subtract them, it is going to be equal to 5.11 years. Now do both of these work? Well, if we go back to our problem statement, we know that T is between 0 and 60. So that means T must be equal to 5.11 years when our area of the reserve is 50 square kilometers. So we have a value for T. Now that we have a value for a T, we can differentiate our function A with respect to T and then substitute this value of T 5.11 years. No Here next we can say then that we already know that A of T is equal to A of T is equal to -160 of T2 + 4 T + 30. So when we differentiate that by using the poor rule, the derivative of -160 of T squared is going to be -1/30 of T, and the derivative of 4 T is going to be 4. And the derivative of or constant 30 is going to be zero. So the derivative of A with respect to T is -130 T plus 4. Finally, we can find the derivative at T equals 5.11 years. So at that value, then our derivative of A with respect to T equals -130 multiplied by 5.11 plus 4. Which equals 3.83 square kilometers per year. So what does this mean? Well, this tells us that the rate of change of the area when it is 50 square kilometers is going to be 3.83 square kilometers per year. If we look back at our answer twices, that means D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.