The length x of a rectangle is decreasing at the rate of 5 cm/min and…

The correct answer is Option (4) → (A) and (C) only

Here \(\frac{dx}{dt} = -5\) (decreasing) and \(\frac{dy}{dt} = 4\) (increasing). Differentiating each measure with respect to t:

Perimeter P = 2(x + y):

    \( \displaystyle \frac{dP}{dt} = 2\left(\frac{dx}{dt} + \frac{dy}{dt}\right) = 2(-5 + 4) = -2 \)

Area A = xy, by the product rule:

    \( \displaystyle \frac{dA}{dt} = x\frac{dy}{dt} + y\frac{dx}{dt} \)

    \( = 8 \times 4 + 6 \times (-5) = 32 - 30 = 2 \)

Diagonal D, with \(D^{2} = x^{2} + y^{2}\), so D = 10 at this instant:

    \( \displaystyle 2D\frac{dD}{dt} = 2x\frac{dx}{dt} + 2y\frac{dy}{dt} \)

    \( \displaystyle \frac{dD}{dt} = \frac{8 \times (-5) + 6 \times 4}{10} = \frac{-16}{10} = -1.6 \)

So the perimeter is decreasing at 2 cm/min, the area is increasing at 2 cm²/min and the diagonal is decreasing at 1.6 cm/min.

Checking the statements

(A) The perimeter is decreasing at the rate of 2 cm/min. — Correct. \(\frac{dP}{dt} = 2(-5 + 4) = -2\).

(B) The area is decreasing at the rate of 2 cm²/min. — Incorrect. \(\frac{dA}{dt} = 32 - 30 = +2\), so the area is increasing.

(C) The length of the diagonal is decreasing at the rate of 1.6 cm/min. — Correct. \(\frac{dD}{dt} = \frac{-40 + 24}{10} = -1.6\).

(D) The perimeter is increasing at the rate of 18 cm/min. — Incorrect. The perimeter is decreasing at 2 cm/min; 18 comes from adding 5 and 4 without the signs.