Question

The base of a triangle is decreasing at a constant rate of 0.2 cm/sec, and the height is increasing at 0.1 cm/sec. If the area is increasing, which answer best describes the constraints on the height ℎ h at the instant when the base is 3 centimeters? a) h>3 b) h<1 c) h>1.5 d) ℎh<1.5

Answer 1

The constraint on the height h of a triangle for the area to increase, given that the base is decreasing at 0.2 cm/sec and the height is increasing at 0.1 cm/sec, is h > 1.5 cm when the base is 3 cm. This is calculated using the derivative of the area formula in terms of time.

Step-by-step explanation:

The student's question involves the rate of change of the base and height of a triangle and its effect on the area of the triangle. The area of the triangle is given by the formula 1/2 × base × height. To find out the constraint on the height (h) when the base is decreasing and the height is increasing such that the area is still increasing, we can use the concept of derivatives from calculus.

For the area (A) of the triangle to increase, the rate of change of the area with respect to time (dA/dt) must be positive. So, if the base is b and the height is h, and their rates of change are given by db/dt = -0.2 cm/sec (decreasing) and dh/dt = 0.1 cm/sec (increasing), respectively, we will use the product rule for differentiation to find dA/dt.

Using the area formula A = 1/2 × b × h, we differentiate both sides with respect to time to obtain:
dA/dt = 1/2 × (b × dh/dt + h × db/dt).

Substitute the given rates and the base b = 3 cm into the derivative to find the constraint on h:

For the area to increase, dA/dt must be greater than 0, thus:
0 < 1/2 × 0.3 cm²/sec - 1/10 × h cm²/sec
0.15 cm²/sec < 1/10 × h cm²/sec
h > 1.5 cm

Therefore, the constraint on the height at that instant is h > 1.5, which corresponds to option (c) h > 1.5.

Answer 2

The constraints on the height of the triangle at the instant when the base is 3 centimeters are h > -1.5 and h > 0. However, for practical purposes, the height cannot be less than 0, so the correct constraint is h > 0.

Step-by-step explanation:

To determine the constraints on the height of the triangle at the instant when the base is 3 centimeters, we need to consider the relationship between the base, height, and area of a triangle. The formula for the area of a triangle is A = (1/2) * base * height. Since the base is decreasing at a constant rate and the height is increasing at a constant rate, we can write the equation:

A' = (1/2) * (base' * height + base * height').

Since the area is increasing, A' > 0. Substituting the given rates of change, we have:

(1/2) * (0.2 * h + 3 * 0.1) > 0.

After simplifying the expression, we get:

0.1h + 0.15 > 0.

Solving for h:

0.1h > -0.15

h > -0.15 / 0.1

h > -1.5.

Since the height of a triangle cannot be negative, the constraint on the height ℎ h at the instant when the base is 3 centimeters is h > -1.5. However, for practical purposes, the height cannot be less than 0. Therefore, we can conclude that the correct constraint on the height is h > 0.