A rectangle is 2 feet by 15 inches. Its length is decreasing by 3 inches/minute and its width is increasing at 4 inches/minute. How fast is the

a) perimeter changing
b) area changing

let the length be x inches

let the width by y inches

A = xy
dA/dt = x dy/dt + y dx/dt
for the given data...
dA/dt = 24(-3) + 15(4) = -12 inches^2/min
the area is decreasing at 12 square inches/min

P = 2x + 2y
dP/dt = 2 dx/dt + 2 dy/dt
= 2(-3) + 2(4) = 2 inches/min

a) Hmm, let's see. The perimeter of a rectangle is given by the formula P = 2L + 2W, where L is the length and W is the width. Now we need to find the rates of change of both L and W, right? Sure thing!

For the length, we know it's decreasing at a rate of 3 inches/minute. So the rate of change for the length would be -3 inches/minute. As for the width, it's increasing at a rate of 4 inches/minute, so the rate of change for the width would be +4 inches/minute.

Now, let's plug in these values into the formula for the perimeter and see what we get. Assuming we're at some specific point in time:

P = 2L + 2W
P = 2(2 feet) + 2(15 inches)
P = 4 feet + 30 inches

To make things consistent, let's convert everything to inches:
P = 48 inches + 30 inches
P = 78 inches

Since we're interested in finding the how fast the perimeter is changing, we can use the chain rule of differentiation. Taking the derivative with respect to time, we get:

dP/dt = 2(dL/dt) + 2(dW/dt)
dP/dt = 2(-3 in/min) + 2(4 in/min)
dP/dt = -6 in/min + 8 in/min
dP/dt = 2 in/min

So, the perimeter is changing at a rate of 2 inches/minute.

b) To find the rate of change of the area, we can use the formula A = LW, where A is the area, L is the length, and W is the width. Since both the length and width are changing with respect to time, we'll need to take their individual derivatives. Let's do some math magic:

A = (2 feet)(15 inches)
A = (24 inches)(15 inches)
A = 360 square inches

Now, let's find the rates of change for both L and W, shall we? We already know that dL/dt = -3 inches/minute and dW/dt = 4 inches/minute.

To find dA/dt (how fast the area is changing):

dA/dt = (dL/dt)(W) + (dW/dt)(L)
dA/dt = (-3 in/min)(15 inches) + (4 in/min)(24 inches)
dA/dt = -45 square inches/minute + 96 square inches/minute
dA/dt = 51 square inches/minute

So, the area of the rectangle is changing at a rate of 51 square inches/minute. Voila!

To find the rate of change of the perimeter and area of the rectangle, we can use the chain rule from calculus. Define the length of the rectangle as L(t) and the width as W(t) at time t, where t is measured in minutes.

Given:
- Change in length: dL/dt = -3 inches/minute (decreasing by 3 inches/minute)
- Change in width: dW/dt = 4 inches/minute (increasing by 4 inches/minute)

We also know the formulas for the perimeter and area of a rectangle:

Perimeter P = 2L + 2W
Area A = L * W

a) To find how fast the perimeter is changing, we need to find dP/dt.
Using the chain rule, we can find:
dP/dt = d(2L)/dt + d(2W)/dt

d(2L)/dt represents the change in the length component of the perimeter, and d(2W)/dt represents the change in the width component.

Substituting the given values:
dP/dt = 2 * dL/dt + 2 * dW/dt
dP/dt = 2(-3) + 2(4)
dP/dt = -6 + 8
dP/dt = 2 inches/minute

Therefore, the perimeter is changing at a rate of 2 inches/minute.

b) To find how fast the area is changing, we need to find dA/dt.
Using the product rule, we can find:
dA/dt = L * dW/dt + W * dL/dt

Substituting the given values:
dA/dt = L * (4) + W * (-3)
dA/dt = 4L - 3W

To find the values of L and W at any given time, we need additional information or initial conditions. Without that, we cannot determine the specific rate of change for the area, but we can still describe the relationship.

If we assume that the rectangle starts as 2 feet by 15 inches, we can convert the length to inches (since the width is already given in inches):
L = 2 ft * 12 in/ft = 24 inches
W = 15 inches

Substituting these values:
dA/dt = 4(24) - 3(15)
dA/dt = 96 - 45
dA/dt = 51 square inches per minute

Therefore, if the rectangle starts as 2 feet by 15 inches, the area will increase at a rate of 51 square inches per minute.

To find the rate of change of the perimeter and area of the rectangle, we need to use calculus and the chain rule. Let's first define our variables:

L = length of the rectangle
W = width of the rectangle

Given information:
dL/dt = -3 inches/minute (the length is decreasing at 3 inches per minute)
dW/dt = 4 inches/minute (the width is increasing at 4 inches per minute)

The perimeter of a rectangle is given by:
P = 2L + 2W

a) To find the rate of change of the perimeter (dP/dt), we need to differentiate the perimeter equation with respect to time (t):
dP/dt = d(2L)/dt + d(2W)/dt

Since L and W are both functions of time, we need to apply the chain rule:

d(2L)/dt = 2 * dL/dt
d(2W)/dt = 2 * dW/dt

Substituting the given values for dL/dt and dW/dt, we have:
dP/dt = 2 * (-3) + 2 * 4

Simplifying:
dP/dt = -6 + 8 = 2 inches/minute

Therefore, the perimeter of the rectangle is changing at a rate of 2 inches per minute.

b) The area of a rectangle is given by:
A = L * W

To find the rate of change of the area (dA/dt), we again need to differentiate the area equation with respect to time:

dA/dt = d(L * W)/dt

Applying the product rule of differentiation, we have:

dA/dt = L * dW/dt + W * dL/dt

Substituting the given values for dW/dt and dL/dt, we have:

dA/dt = L * 4 + W * (-3)

Since L and W are given, we can substitute their values into the equation:

dA/dt = 2 ft * 4 inches/minute + 15 inches * (-3 inches/minute)

Converting feet to inches (1 ft = 12 inches), we have:

dA/dt = 2 * 12 inches * 4 inches/minute + 15 inches * (-3 inches/minute)

Simplifying:

dA/dt = 96 inches/min - 45 inches/min = 51 inches/min

Therefore, the area of the rectangle is changing at a rate of 51 square inches per minute.