The length l of a rectangle is decrasing at a rate of 5 cm/sec while the width w is increasing at a rate of 2 cm/sec. When l = 15 cm and w = 7 cm, find the following rates of change:
The rate of change of the area:
Answer = cm^2/sec.
The rate of change of the perimeter:
Answer = cm/sec.
The rate of change of the diagonals:
Answer = cm/sec.
To find the rate of change of the area, perimeter, and diagonals, we'll first need to determine the formulas for those quantities in terms of the length and width of the rectangle.
1. Rate of Change of Area:
The area of a rectangle is given by the formula A = l * w. Since both length and width are changing, we need to use the product rule of differentiation to find the rate of change. Let's differentiate the area formula with respect to time (t):
dA/dt = (dl/dt) * w + (dw/dt) * l
Given that dl/dt = -5 cm/sec (decreasing at a rate of 5 cm/sec) and dw/dt = 2 cm/sec (increasing at a rate of 2 cm/sec), and substituting the values of l = 15 cm and w = 7 cm, we can calculate the rate of change of the area:
dA/dt = (-5) * 7 + 2 * 15
= -35 + 30
= -5 cm^2/sec
Therefore, the rate of change of the area is -5 cm^2/sec.
2. Rate of Change of Perimeter:
The perimeter of a rectangle is given by the formula P = 2 * (l + w). Again, since both length and width are changing, we'll use the product rule to find the derivative:
dP/dt = 2 * ((dl/dt) + (dw/dt))
Substituting the values of dl/dt = -5 cm/sec and dw/dt = 2 cm/sec, we can calculate the rate of change of the perimeter:
dP/dt = 2 * ((-5) + 2)
= 2 * (-3)
= -6 cm/sec
Therefore, the rate of change of the perimeter is -6 cm/sec.
3. Rate of Change of Diagonals:
The diagonals of a rectangle are given by the formula D = sqrt(l^2 + w^2). Differentiating this formula with respect to time:
dD/dt = (1/2) * (l^2 + w^2)^(-1/2) * (2l * dl/dt + 2w * dw/dt)
Let's substitute the values of l = 15 cm, w = 7 cm, dl/dt = -5 cm/sec, and dw/dt = 2 cm/sec to calculate the rate of change of the diagonals:
dD/dt = (1/2) * (15^2 + 7^2)^(-1/2) * (2 * 15 * (-5) + 2 * 7 * 2)
= (1/2) * (225 + 49)^(-1/2) * (-150 + 28)
= (1/2) * (274)^(-1/2) * (-122)
≈ -4.44 cm/sec
Therefore, the rate of change of the diagonals is approximately -4.44 cm/sec.