Answer to Question #148149 in Calculus for Sean

Question

Two parallel sides of a rectangle are increasing in length at the rate of 2 (in/sec) while the other two sides are decreasing in length such that the figure remains a rectangle with a constant area of 50 𝑖𝑛^2. What is the rate of change of the perimeter when the length of the increasing pair of sides is 5 𝑖𝑛?

Answer 1

The question is incomplete, so I will try to solve it by taking some assumptions,

because without adding some assumption values it is impossible to solve this problem.

so lets take we are asked to find the rate of change of the perimeter P and the increasing side is equal to 5 in,

lets solve for this case; the increase rate of the side is 2 in/sec, and the area S=50 in²

x and y are two sides of the rectangle, the area equals to the product of two sides

S=xy, we have x=5 from here we can find y=10 in, and P=2x+2y;

P is changing with respect to time its derivative is dP/dt=2dx/dt+2dy/dt ,

the derivative of S is zero because area is constant dS/dt=0;

and dS/dt=x*dy/dt+y*dx/dt=0, x=5, y=10. the increase rate of x is dx/dt and it is equal to dx/dt=2 in/sec

5dy/dt+10dx/dt=0, from here we can find 5dy/dt+10(2)=0, dy/dt=-4;

now, we can find the rate of change of perimeter dP/dt=2*(2)+2*(-4)= -4in/sec;

the answer is -4in/sec.