Q.A spherical ball of salt is dissolving in water in such a manner that the rate of decrease of the volume at any instant is proportional to the surface. Prove that the radius is decreasing at a constant rate.

Showing the 12 most recent of 17 on this concept.

1. COMEDK 2023Set 2023-E1 markMCQ
   Q.If the volume of a sphere is increasing at a constant rate, then the rate at which its radius is increasing is (A) inversely proportional to its surface area (B) proportional to the radius (C) a constant (D) inversely proportional to the radius

   ›Reveal solutionSolution

   i.e. the rate of increase of the radius is inversely proportional to the surface area (equivalently, inversely proportional to r^2 - NOT simply inversely proportional to r, so (D) is wrong).

   Concept: related rates for a sphere.

   V = (4/3) pi r^3

   dV/dt = 4 pi r^2 * dr/dt

   Given dV/dt = k (a constant), so

   dr/dt = k / (4 pi r^2).

   But 4 pi r^2 is exactly the surface area S of the sphere. Therefore

   dr/dt = k / S,

   i.e. the rate of increase of the radius is inversely proportional to the surface area (equivalently, inversely proportional to r^2 - NOT simply inversely proportional to r, so (D) is wrong).

   ✓Final answer

   The correct option is (A) — inversely proportional to its surface area

   ANSWER: A

2. COMEDK 2025Set 2025-M1 markMCQ
   Q.A spherical snowball is melting such that its volume is decreasing at the rate of $1 \mathrm{~cm}^3 / \mathrm{min}$. The rate at which the diameter is decreasing when the diameter is 10 cm is (A) $\frac{11}{75 \pi} \mathrm{~cm} / \mathrm{min}$ (B) $\frac{1}{50 \pi} \mathrm{~cm} / \mathrm{min}$ (C) $\frac{2}{75 \pi} \mathrm{~cm} / \mathrm{min}$ (D) $\frac{1}{25 \pi} \mathrm{~cm} / \mathrm{min}$

   ›Reveal solutionSolution

   We relate the rate of change of volume to the rate of change of diameter using the formula for the volume of a sphere and implicit differentiation. The diameter decreases at $\frac{1}{50\pi}$ cm/min when the diameter is 10 cm, so the correct option is (B).

   ---

   Concept and intuition:

   This is a classic related rates problem. The snowball’s volume shrinks at a known constant rate, and we want how fast its diameter shrinks at a particular instant. The key is to connect volume $V$ and diameter $D$ through the sphere’s volume formula, then differentiate both sides with respect to time $t$. Because we know $\frac{dV}{dt}$ and want $\frac{dD}{dt}$, we just substitute the given diameter and solve.

   ---

   Step-by-step solution:

   1. Write the volume in terms of diameter. The volume of a sphere of radius $r$ is $V = \frac{4}{3}\pi r^3$. Since the diameter $D = 2r$, we have $r = D/2$. Substituting:

   $V = \frac{4}{3}\pi \left(\frac{D}{2}\right)^3 = \frac{4}{3}\pi \cdot \frac{D^3}{8} = \frac{\pi}{6} D^3.$

   This expresses $V$ directly as a function of $D$, which is convenient because we want $\frac{dD}{dt}$.

   2. Differentiate with respect to time $t$. Both $V$ and $D$ depend on $t$, so we use implicit differentiation:

   $\frac{dV}{dt} = \frac{\pi}{6} \cdot 3D^2 \cdot \frac{dD}{dt} = \frac{\pi}{2} D^2 \frac{dD}{dt}.$

   3. Plug in known values. We are told $\frac{dV}{dt} = -1$ cm³/min (negative because volume is decreasing). At the instant of interest, $D = 10$ cm. Substitute:

   $-1 = \frac{\pi}{2} (10)^2 \cdot \frac{dD}{dt} = \frac{\pi}{2} \cdot 100 \cdot \frac{dD}{dt} = 50\pi \cdot \frac{dD}{dt}.$

   4. Solve for $\frac{dD}{dt}$.

   $\frac{dD}{dt} = \frac{-1}{50\pi} \ \text{cm/min}.$

   The negative sign confirms the diameter is decreasing. The question asks for the rate at which it is decreasing, so we report the magnitude: $\frac{1}{50\pi}$ cm/min.

   Tip

   A common mistake is to use the radius instead of the diameter in the formula, or to forget the factor of 2 when converting between $dr/dt$ and $dD/dt$. Writing $V$ directly in terms of $D$ avoids that pitfall entirely.

   Watch out

   Do not forget the negative sign when interpreting the rate. The problem asks for “the rate at which the diameter is decreasing” — that is the absolute value of $\frac{dD}{dt}$, so it is positive.

   ---

   ✓Final answer

   The correct option is (B).

   ANSWER: B

3. COMEDK 2024Set 2024-E1 markMCQ
   Q.The side of a cube is equal to the diameter of a sphere. If the side and radius increase at the same rate then the ratio of the increase of their surface area is (A) $3: \pi$ (B) $\pi: 6$ (C) $2 \pi: 3$ (D) $3: 2 \pi$

   ›Reveal solutionSolution

   The problem asks for the ratio of the rates of increase of surface areas of a cube and a sphere when their side and radius increase at the same rate, given the side equals the sphere’s diameter. The answer is $3 : \pi$, which corresponds to option (A).

   We start by understanding the relationship: the cube’s side length $s$ equals the sphere’s diameter, so $s = 2r$, where $r$ is the sphere’s radius. Both $s$ and $r$ increase at the same rate, meaning $\frac{ds}{dt} = \frac{dr}{dt}$. We want the ratio of the rates of change of their surface areas.

   Concept and intuition:

   Surface area growth depends on both the current size and the rate of change of the linear dimension. Since the cube and sphere have different formulas for surface area, their rates of change will differ even when their linear dimensions grow at the same speed. The key is to differentiate each surface area with respect to time, then substitute the given relationship $s = 2r$ and the equal rate condition.

   ---

   1. Write the surface area formulas.

      • Cube surface area: $A_{\text{cube}} = 6s^2$
      • Sphere surface area: $A_{\text{sphere}} = 4\pi r^2$

   2. Differentiate both with respect to time $t$.

      • $\frac{dA_{\text{cube}}}{dt} = 12s \cdot \frac{ds}{dt}$
      • $\frac{dA_{\text{sphere}}}{dt} = 8\pi r \cdot \frac{dr}{dt}$

   3. Apply the given conditions.

      • The side equals the diameter: $s = 2r$.
      • The rates are equal: $\frac{ds}{dt} = \frac{dr}{dt}$. Substitute $s = 2r$ into the cube’s rate:

   $\frac{dA_{\text{cube}}}{dt} = 12(2r) \cdot \frac{dr}{dt} = 24r \cdot \frac{dr}{dt}$

   4. Form the ratio of the rates.

   $\frac{\frac{dA_{\text{cube}}}{dt}}{\frac{dA_{\text{sphere}}}{dt}} = \frac{24r \cdot \frac{dr}{dt}}{8\pi r \cdot \frac{dr}{dt}} = \frac{24}{8\pi} = \frac{3}{\pi}$

   So the ratio (cube : sphere) is $3 : \pi$.

   Tip

   A common mistake is to forget that the cube’s side equals the diameter, not the radius. Using $s = r$ instead of $s = 2r$ would give a different ratio $\pi : 6$, which is option (B) — a tempting distractor.

   ✓Final answer

   The correct option is (A).

   ANSWER: A

4. COMEDK 2026Set 2026-M1 markMCQ
   Q.A square plate is contracting at a uniform rate of $2 \mathrm{~cm}^2 / \mathrm{min}$. The rate at which the perimeter is decreasing when the side of the square is 16 cm is: (A) $\frac{1}{8} \mathrm{~cm} / \mathrm{min}$ (B) $\frac{1}{4} \mathrm{~cm} / \mathrm{min}$ (C) $16 \mathrm{~cm} / \mathrm{min}$ (D) $32 \mathrm{~cm} / \mathrm{min}$

   ›Reveal solutionSolution

   The area decreases at a constant rate; we relate the side length to area, differentiate with respect to time, and then find the perimeter’s rate of change. The perimeter decreases at $\frac{1}{4}$ cm/min when the side is 16 cm.

   We have a square plate whose area is shrinking at a steady rate of $2\ \text{cm}^2/\text{min}$. The question asks: at the moment the side length is 16 cm, how fast is the perimeter decreasing?

   The key idea is related rates: we connect the changing area to the changing side length, then connect the side length to the perimeter. Because the contraction is uniform, the side length shrinks at a rate that depends on the current side length.

   ---

   1. Define variables and given rate Let $s$ be the side length (in cm) and $A$ the area (in cm²). For a square:

   $A = s^2$

   We are told:

   $\frac{dA}{dt} = -2 \quad (\text{negative because area is decreasing})$

   2. Relate the rates of area and side Differentiate $A = s^2$ with respect to time $t$:

   $\frac{dA}{dt} = 2s \cdot \frac{ds}{dt}$

   Substitute the known rate:

   $-2 = 2s \cdot \frac{ds}{dt}$

   Solve for $\frac{ds}{dt}$:

   $\frac{ds}{dt} = -\frac{1}{s}$

   At the instant $s = 16$ cm:

   $\frac{ds}{dt} = -\frac{1}{16}\ \text{cm/min}$

   The negative sign confirms the side length is decreasing.

   3. Find the perimeter’s rate of change Perimeter $P = 4s$. Differentiate:

   $\frac{dP}{dt} = 4 \cdot \frac{ds}{dt}$

   Plug in $\frac{ds}{dt} = -\frac{1}{16}$:

   $\frac{dP}{dt} = 4 \times \left(-\frac{1}{16}\right) = -\frac{1}{4}\ \text{cm/min}$

   The negative sign means the perimeter is decreasing. The question asks for the rate at which it is decreasing, so we take the magnitude: $\frac{1}{4}$ cm/min.

   Watch out

   A common mistake is to forget that $\frac{ds}{dt}$ depends on $s$ — it is not constant. If you incorrectly assume $\frac{ds}{dt}$ is constant, you might get a wrong answer like 16 or 32 cm/min.

   Tip

   Notice that $\frac{ds}{dt} = -\frac{1}{s}$ means the side shrinks faster when the square is smaller — intuitive, because a fixed area loss is a larger fraction of a small square.

   ✓Final answer

   The correct option is (B).

   ANSWER: B

5. COMEDK 2025Set 2025-M1 markMCQ
   Q.Oil from a conical funnel is dripping at the rate of $5 \mathrm{~cm}^3 / \mathrm{s}$. If the radius and height of the funnel are 10 cm and 20 cm respectively, then the rate at which the oil level drops when it is 5 cm from the top is (A) $\frac{8}{45 \pi} \mathrm{~cm} / \mathrm{s}$ (B) $-\frac{2 \pi}{45} \mathrm{~cm} / \mathrm{s}$ (C) $-\frac{4 \pi}{45} \mathrm{~cm} / \mathrm{s}$ (D) $-\frac{4}{45 \pi} \mathrm{~cm} / \mathrm{s}$

   ›Reveal solutionSolution

   The rate at which the oil level drops is found by relating the volume of a cone to its height using similar triangles, then differentiating with respect to time. The answer is $-\frac{4}{45\pi} \text{ cm/s}$, which corresponds to option (D).

   ---

   Concept & Intuition

   This is a classic related rates problem. Oil is draining from a conical funnel, so the volume is decreasing at a known rate ($dV/dt = -5$ cm³/s). We want the rate at which the height of the oil changes ($dh/dt$) when the oil is at a particular depth. The key twist: as the oil level drops, the radius of the oil's surface also shrinks, because the funnel is conical. The radius and height of the oil are not independent — they are linked by the geometry of the cone (similar triangles). So we first express volume purely in terms of height, then differentiate.

   ---

   Step-by-step solution

   1. Set up the geometry. The funnel is a right circular cone with radius $R = 10$ cm and height $H = 20$ cm. At any moment, the oil forms a smaller cone of height $h$ (measured from the tip) and radius $r$. By similar triangles:

   $\frac{r}{h} = \frac{R}{H} = \frac{10}{20} = \frac{1}{2}$

   So $r = \frac{h}{2}$.

   2. Write the volume of oil in terms of $h$. Volume of a cone: $V = \frac{1}{3}\pi r^2 h$. Substitute $r = h/2$:

   $V = \frac{1}{3}\pi \left(\frac{h}{2}\right)^2 h = \frac{1}{3}\pi \cdot \frac{h^2}{4} \cdot h = \frac{\pi}{12} h^3$

   3. Differentiate with respect to time $t$. Using the chain rule:

   $\frac{dV}{dt} = \frac{\pi}{12} \cdot 3h^2 \cdot \frac{dh}{dt} = \frac{\pi}{4} h^2 \frac{dh}{dt}$

   4. Plug in known values. We are told the oil is dripping out at $5$ cm³/s, so $dV/dt = -5$ (negative because volume is decreasing). The oil is 5 cm from the top of the funnel. Since the funnel is 20 cm tall, the oil height from the tip is $h = 20 - 5 = 15$ cm. Substitute:

   $-5 = \frac{\pi}{4} (15)^2 \frac{dh}{dt}$

   $-5 = \frac{\pi}{4} \cdot 225 \cdot \frac{dh}{dt}$

   $-5 = \frac{225\pi}{4} \frac{dh}{dt}$

   5. Solve for $dh/dt$.

   $\frac{dh}{dt} = -5 \cdot \frac{4}{225\pi} = -\frac{20}{225\pi} = -\frac{4}{45\pi}$

   The negative sign confirms the oil level is dropping.

   Watch out

   A common mistake is to use the full funnel's radius and height directly in the volume formula, forgetting that the oil's radius changes with height. Always use similar triangles to relate $r$ and $h$ for the current oil cone.

   Tip

   Notice that the rate $dh/dt$ is not constant — it depends on $h$. That’s why we needed the specific height (15 cm) to get a numerical answer.

   ---

   ✓Final answer

   The correct option is (D).

   ANSWER: D

6. COMEDK 2025Set 2025-E1 markMCQ
   Q.If the length of the diagonal of a square is increasing at the rate of $0.1 \mathrm{~cm} / \mathrm{sec}$. What is the rate of increase of its area when the side is $\frac{15}{\sqrt{2}} \mathrm{~cm}$ ? (A) $3 \mathrm{~cm}^2 / \mathrm{sec}$ (B) $0.15 \mathrm{~cm}^2 / \mathrm{sec}$ (C) $1.5 \mathrm{~cm}^2 / \mathrm{sec}$ (D) $3 \sqrt{2} \mathrm{~cm}^2 / \mathrm{sec}$

   ›Reveal solutionSolution

   The key idea is to relate the side length and diagonal of a square, then use the chain rule to connect the rates of change of the diagonal and the area. The area increases at $1.5 \, \text{cm}^2/\text{sec}$ when the side is $\frac{15}{\sqrt{2}}$ cm, so the correct option is (C).

   We are told the diagonal of a square is increasing at a constant rate of $0.1 \, \text{cm/s}$. We need the rate of increase of the area when the side length is $\frac{15}{\sqrt{2}} \, \text{cm}$. The natural approach is to express the area in terms of the diagonal, then differentiate with respect to time.

   Concept and intuition:

   For a square, the diagonal $d$ and side $s$ are related by $d = s\sqrt{2}$. The area $A = s^2$. If we know how fast $d$ changes, we can find how fast $s$ changes, and then how fast $A$ changes. Alternatively, we can directly relate $A$ to $d$: since $s = d/\sqrt{2}$, then $A = (d/\sqrt{2})^2 = d^2/2$. Differentiating this gives $dA/dt = d \cdot (dd/dt)$. This is simpler because we don't need to find $ds/dt$ separately.

   Let's work through step by step.

   1. Relate area to diagonal. For a square with side $s$, diagonal $d = s\sqrt{2}$ and area $A = s^2$. Substituting $s = d/\sqrt{2}$ gives:

   $A = \left(\frac{d}{\sqrt{2}}\right)^2 = \frac{d^2}{2}.$

   2. Differentiate with respect to time. Using the chain rule:

   $\frac{dA}{dt} = \frac{d}{dt}\left(\frac{d^2}{2}\right) = \frac{1}{2} \cdot 2d \cdot \frac{dd}{dt} = d \cdot \frac{dd}{dt}.$

   We are given $\frac{dd}{dt} = 0.1 \, \text{cm/s}$.

   3. Find the diagonal when the side is $\frac{15}{\sqrt{2}}$ cm. Since $d = s\sqrt{2}$,

   $d = \frac{15}{\sqrt{2}} \cdot \sqrt{2} = 15 \, \text{cm}.$

   4. Plug into the rate equation.

   $\frac{dA}{dt} = 15 \times 0.1 = 1.5 \, \text{cm}^2/\text{sec}.$

   Tip

   Notice we never needed to compute $ds/dt$ explicitly. By expressing area directly in terms of the diagonal, we saved a step and reduced the chance of algebraic error.

   Watch out

   A common mistake is to forget that the diagonal and side are not independent — students sometimes try to differentiate $A = s^2$ without first relating $s$ to $d$, leading to confusion about which rate is given.

   Thus, the rate of increase of the area is $1.5 \, \text{cm}^2/\text{sec}$.

   ✓Final answer

   The correct option is (C).

   ANSWER: C

7. COMEDK 2026Set 2026-A1 markMCQ
   Q.An open hemispherical storage tank has radius 13 m . Oil flows into the tank such that the depth ' $\boldsymbol{h}$ ' of oil in the tank changes at the rate of $3 \mathrm{~m} / \mathrm{hr}$. When the depth $\boldsymbol{h}=1 \mathrm{~m}$, the rate of change of the area of the top surface of the oil is (A) $72 \pi \mathrm{~m}^2 / \mathrm{hr}$ (B) $75 \pi \mathrm{~m}^2 / \mathrm{hr}$ (C) $24 \pi \mathrm{~m}^2 / \mathrm{hr}$ (D) $26 \pi \mathrm{~m}^2 / \mathrm{hr}$

   ›Reveal solutionSolution

   The oil surface is a circle of radius $r$ with $r^2=2Rh-h^2$ for a bowl of radius $R$. So area $A=\pi(2Rh-h^2)$ and $\frac{dA}{dt}=\pi(2R-2h)\frac{dh}{dt}$. At $R=13,\ h=1,\ \frac{dh}{dt}=3$ this is $72\pi\ \text{m}^2/\text{hr}$ — option (A).

   Concept & Intuition

   In a hemispherical bowl of radius $R$, the free surface at oil depth $h$ (measured from the lowest point) is a horizontal circle. Its radius $r$ comes from the sphere geometry: the surface is a distance $R-h$ from the centre, so $r^2=R^2-(R-h)^2=2Rh-h^2$. Differentiate the surface area with respect to time and use the given $\frac{dh}{dt}$.

   Step-by-step solution

   1. Radius of the top circle at depth $h$:

   $r^2=R^2-(R-h)^2=2Rh-h^2.$

   2. Area of the top surface:

   $A=\pi r^2=\pi\left(2Rh-h^2\right).$

   3. Differentiate with respect to time:

   $\frac{dA}{dt}=\pi\left(2R-2h\right)\frac{dh}{dt}.$

   4. Substitute $R=13$, $h=1$, $\dfrac{dh}{dt}=3$:

   $\frac{dA}{dt}=\pi\,(2\cdot13-2\cdot1)\,(3)=\pi\,(26-2)\,(3)=\pi\,(24)(3)=72\pi.$

   ✓Final answer

   $\dfrac{dA}{dt}=72\pi\ \text{m}^2/\text{hr}$ — option (A).

   ANSWER: A

8. COMEDK 2023Set 2023-E1 markMCQ
   Q.The altitude of a cone is $20 \mathrm{~cm}$ and its semi vertical angle is $30^{\circ}$. If the semi vertical angle is increasing at the rate of $2^0$ per second, then the radius of the base is increasing at the rate of (A) $160 \mathrm{~cm} / \mathrm{sec}$ (B) $10 \mathrm{~cm} / \mathrm{sec}$ (C) $\frac{160}{3} \mathrm{~cm} / \mathrm{sec}$ (D) $30 \mathrm{~cm} / \mathrm{sec}$

   ›Reveal solutionSolution

   (Note on units: the paper quotes the angular rate as '2 degrees per second' but the options are only consistent with treating that rate as 2 in the same angular unit used for the derivative - i.e. the option set is built on dr/dt = h sec^2 a * 2 = 160/3. Converting 2 degrees to radians would give 20*(4/3)*(pi/90) ~ 0.93 cm/s, which matches none of the four options. The intended and only available answer is 160/3 cm/sec.)

   Concept: related rates on a cone. With altitude h fixed and semi-vertical angle alpha varying, r = h tan alpha, so dr/dt = h sec^2(alpha) * d(alpha)/dt.

   Given h = 20 cm, alpha = 30 degrees, d(alpha)/dt = 2 per second.

   sec^2(30) = 1/cos^2(30) = 1/(3/4) = 4/3.

   dr/dt = 20 * (4/3) * 2 = 160/3 cm/sec.

   (Note on units: the paper quotes the angular rate as '2 degrees per second' but the options are only consistent with treating that rate as 2 in the same angular unit used for the derivative - i.e. the option set is built on dr/dt = h sec^2 a * 2 = 160/3. Converting 2 degrees to radians would give 20*(4/3)*(pi/90) ~ 0.93 cm/s, which matches none of the four options. The intended and only available answer is 160/3 cm/sec.)

   ✓Final answer

   The correct option is (C) — $\frac{160}{3} \mathrm{~cm} / \mathrm{sec}$

   ANSWER: C

9. COMEDK 2024Set 2024-M1 markMCQ
   Q.For a given curve $y=2 x-x^2$, when $x$ increases at the rate of 3 units/sec, then how does the slope of the curve change? (A) Decreasing at 3 units/sec (B) Increasing at 3 units/sec (C) Decreasing at 6 units/sec (D) Increasing at 6 units/sec

   ›Reveal solutionSolution

   The slope of the curve is given by the derivative $\frac{dy}{dx} = 2 - 2x$, and its rate of change with respect to time is $\frac{d}{dt}(\text{slope}) = -2 \cdot \frac{dx}{dt}$. With $\frac{dx}{dt} = 3$ units/sec, the slope decreases at 6 units/sec, so the answer is (C).

   ---

   The key idea is that we are not asked for the slope itself, but for how fast the slope changes over time. That means we need the time derivative of the slope, using the chain rule, because the slope depends on $x$, and $x$ itself changes with time.

   Concept & Intuition:

   Imagine a point moving along the parabola $y = 2x - x^2$. As $x$ increases steadily (3 units every second), the slope of the tangent line at the moving point changes. The slope is $m = 2 - 2x$, a linear function of $x$. If $x$ increases, $m$ decreases because of the $-2x$ term. The question is: how fast does $m$ decrease? That’s just the derivative of $m$ with respect to time: $\frac{dm}{dt} = \frac{dm}{dx} \cdot \frac{dx}{dt}$.

   ---

   Step-by-step solution:

   1. Find the slope of the curve as a function of $x$. The slope at any point is the derivative $\frac{dy}{dx}$:

   $y = 2x - x^2 \quad \Rightarrow \quad \frac{dy}{dx} = 2 - 2x.$

   So the slope $m(x) = 2 - 2x$.

   2. We need the rate of change of the slope with respect to time. That is $\frac{dm}{dt}$. Since $m$ depends on $x$, and $x$ depends on $t$, use the chain rule:

   $\frac{dm}{dt} = \frac{dm}{dx} \cdot \frac{dx}{dt}.$

   3. Compute $\frac{dm}{dx}$. From $m = 2 - 2x$, differentiate with respect to $x$:

   $\frac{dm}{dx} = -2.$

   This is constant — the slope of the curve changes linearly with $x$, so its rate of change per unit $x$ is always $-2$.

   4. Use the given $\frac{dx}{dt}$. The problem states $x$ increases at 3 units/sec, so

   $\frac{dx}{dt} = 3.$

   5. Multiply to find $\frac{dm}{dt}$:

   $\frac{dm}{dt} = (-2) \cdot 3 = -6.$

   The negative sign means the slope is decreasing at a rate of 6 units per second.

   Tip

   A common mistake is to confuse the slope itself with its rate of change. The slope at a given $x$ is $2-2x$, but the rate at which that slope changes over time is constant ($-6$ here) because $\frac{dm}{dx}$ is constant and $\frac{dx}{dt}$ is constant. So the slope decreases uniformly.

   Watch out

   Do not plug a specific $x$ value into $2-2x$ and then differentiate that number — that gives zero. The slope changes because $x$ moves; you must use the chain rule.

   ---

   ✓Final answer

   The correct option is (C).

   ANSWER: C

10. COMEDK 2026Set 2026-M1 markMCQ
    Q.If $\mathbf{3 ~ c m} / \mathbf{s}$ is the rate at which the side of an equilateral triangle increases, then the rate of change of area, when the side is $\mathbf{1 2 ~ c m}$ is: (A) $9 \sqrt{3} \mathrm{~cm}^2 / \mathrm{s}$ (B) $18 \mathrm{~cm}^2 / \mathrm{s}$ (C) $6 \sqrt{3} \mathrm{~cm}^2 / \mathrm{s}$ (D) $18 \sqrt{3} \mathrm{~cm}^2 / \mathrm{s}$

    ›Reveal solutionSolution

    The area of an equilateral triangle depends on its side length; by differentiating the area formula with respect to time, we find the rate of change of area when the side is 12 cm is $18\sqrt{3} \, \text{cm}^2/\text{s}$, which corresponds to option (D).

    We are told the side length $s$ of an equilateral triangle increases at a constant rate: $\frac{ds}{dt} = 3$ cm/s. We need the rate of change of the area $A$ when $s = 12$ cm. The key idea is to relate $A$ to $s$ using geometry, then differentiate with respect to time $t$ using the chain rule — this turns a static formula into a dynamic relationship.

    1. Area of an equilateral triangle in terms of its side For an equilateral triangle of side $s$, the height is $\frac{\sqrt{3}}{2}s$ (by splitting it into two 30-60-90 right triangles). The area is

    $A = \frac{1}{2} \cdot \text{base} \cdot \text{height} = \frac{1}{2} \cdot s \cdot \frac{\sqrt{3}}{2}s = \frac{\sqrt{3}}{4} s^2.$

    2. Differentiate with respect to time Both $A$ and $s$ are functions of time $t$. Using the chain rule:

    $\frac{dA}{dt} = \frac{d}{dt}\left( \frac{\sqrt{3}}{4} s^2 \right) = \frac{\sqrt{3}}{4} \cdot 2s \cdot \frac{ds}{dt} = \frac{\sqrt{3}}{2} s \cdot \frac{ds}{dt}.$

    3. Substitute the given values We have $\frac{ds}{dt} = 3$ cm/s and $s = 12$ cm at the moment of interest.

    $\frac{dA}{dt} = \frac{\sqrt{3}}{2} \cdot 12 \cdot 3 = \frac{\sqrt{3}}{2} \cdot 36 = 18\sqrt{3} \, \text{cm}^2/\text{s}.$

    Tip

    Notice that the factor $\frac{\sqrt{3}}{2} s$ is exactly the height of the triangle. So the rate of change of area is simply (height) × (rate of change of side). This geometric shortcut can save time in similar problems.

    Watch out

    A common mistake is forgetting the factor $\frac{\sqrt{3}}{4}$ or misapplying the chain rule — for example, writing $\frac{dA}{dt} = \frac{\sqrt{3}}{2} s$ without multiplying by $\frac{ds}{dt}$. Always check that you’ve differentiated with respect to $t$, not $s$.

    ✓Final answer

    The correct option is (D).

    ANSWER: D

11. COMEDK 2024Set 2024-M1 markMCQ
    Q.The side of an equilateral triangle expands at the rate of $\sqrt{3} \mathrm{~cm} / \mathrm{sec}$. When the side is $12 \mathrm{~cm}$, the rate of increase of its area is (A) $18 \mathrm{~cm}^2 / \mathrm{sec}$ (B) $12 \mathrm{~cm}^2 / \mathrm{sec}$ (C) $10 \mathrm{~cm}^2 / \mathrm{sec}$ (D) $3 \sqrt{3} \mathrm{~cm}^2 / \mathrm{sec}$

    ›Reveal solutionSolution

    The area of an equilateral triangle is $A = \frac{\sqrt{3}}{4} s^2$. Differentiating with respect to time gives $\frac{dA}{dt} = \frac{\sqrt{3}}{2} s \frac{ds}{dt}$. Substituting $s = 12$ cm and $\frac{ds}{dt} = \sqrt{3}$ cm/s yields $\frac{dA}{dt} = 18$ cm²/s, so the correct option is (A).

    Concept & Intuition

    This is a classic related rates problem. The key idea: when a geometric shape changes size, its area changes at a rate that depends on both its current dimensions and how fast those dimensions are changing. Here, the side length grows at a constant speed, but the area grows faster as the side gets longer because area depends on the square of the side. We connect the rates using calculus — specifically, implicit differentiation with respect to time.

    Step-by-step solution

    1. Write the formula for the area of an equilateral triangle. For an equilateral triangle of side $s$, the area is

    $A = \frac{\sqrt{3}}{4} s^2.$

    (Derivation: height $= \frac{\sqrt{3}}{2}s$, so area $= \frac12 \cdot \text{base} \cdot \text{height} = \frac12 \cdot s \cdot \frac{\sqrt{3}}{2}s = \frac{\sqrt{3}}{4}s^2$.)

    2. Differentiate both sides with respect to time $t$. Since $s$ changes with time, $A$ also changes. Using the chain rule:

    $\frac{dA}{dt} = \frac{\sqrt{3}}{4} \cdot 2s \cdot \frac{ds}{dt} = \frac{\sqrt{3}}{2} s \frac{ds}{dt}.$

    This equation tells us the rate of change of area at any instant, given the side length $s$ and its rate of change $\frac{ds}{dt}$.

    3. Plug in the known values. We are given:
       • $\frac{ds}{dt} = \sqrt{3}$ cm/s (the side expands at this rate),
       • $s = 12$ cm (the side length at the moment we care about). Substituting:

    $\frac{dA}{dt} = \frac{\sqrt{3}}{2} \cdot 12 \cdot \sqrt{3} = \frac{\sqrt{3} \cdot 12 \cdot \sqrt{3}}{2} = \frac{12 \cdot 3}{2} = \frac{36}{2} = 18.$

    4. State the units. Since $s$ is in cm and $\frac{ds}{dt}$ in cm/s, $\frac{dA}{dt}$ comes out in cm²/s. So the rate is $18$ cm²/s.

    Watch out

    A common mistake is to forget the factor $\frac{\sqrt{3}}{4}$ or to accidentally use the formula for a square ($s^2$) instead of the equilateral triangle. Always double-check the shape’s area formula.

    Tip

    Notice that $\frac{dA}{dt}$ is proportional to $s$ — so as the triangle grows, the area accelerates even if the side grows at a constant rate. That’s why the answer is larger than you might guess from just multiplying $\frac{ds}{dt}$ by something simple.

    ✓Final answer

    The correct option is (A).

    ANSWER: A

12. COMEDK 2025Set 2025-A1 markMCQ
    Q.$x=a(\theta+\sin \theta)$ and $y=a(1-\cos \theta)$ represents the equation of a curve. If $\theta$ changes at a constant rate $k$ then the rate of change of the slope of the tangent to the curve at $\theta=\frac{\pi}{3}$ is (A) $2 k$ (B) $\frac{k}{3}$ (C) $\frac{2 k}{\sqrt{3}}$ (D) $\frac{2 k}{3}$

    ›Reveal solutionSolution

    The problem asks for the rate of change of the slope of the tangent, not the slope itself. We find $\frac{dy}{dx}$ in terms of $\theta$, then differentiate with respect to time using the chain rule, using $\frac{d\theta}{dt}=k$. At $\theta=\frac{\pi}{3}$, the result simplifies to $\frac{2k}{3}$, so the correct option is (D).

    We are given a cycloid-like parametric curve:

    $x = a(\theta + \sin\theta)$, $y = a(1 - \cos\theta)$, with $\theta$ increasing at constant rate $k$, i.e. $\frac{d\theta}{dt} = k$.

    The slope of the tangent is $\frac{dy}{dx}$. But the question asks for the rate of change of that slope with respect to time — that is $\frac{d}{dt}\left(\frac{dy}{dx}\right)$. This is a classic related-rates problem in parametric form.

    ---

    1. Find the slope in terms of $\theta$

    First compute derivatives with respect to $\theta$:

    $\frac{dx}{d\theta} = a(1 + \cos\theta), \quad \frac{dy}{d\theta} = a\sin\theta$

    Then the slope is:

    $\frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} = \frac{a\sin\theta}{a(1+\cos\theta)} = \frac{\sin\theta}{1+\cos\theta}$

    Using the identity $\sin\theta = 2\sin(\theta/2)\cos(\theta/2)$ and $1+\cos\theta = 2\cos^2(\theta/2)$, this simplifies to:

    $\frac{dy}{dx} = \frac{2\sin(\theta/2)\cos(\theta/2)}{2\cos^2(\theta/2)} = \tan\frac{\theta}{2}$

    So the slope at any $\theta$ is simply $\tan(\theta/2)$.

    ---

    2. Differentiate the slope with respect to time

    We want $\frac{d}{dt}\left(\frac{dy}{dx}\right) = \frac{d}{dt}\left(\tan\frac{\theta}{2}\right)$.

    By the chain rule:

    $\frac{d}{dt}\left(\tan\frac{\theta}{2}\right) = \sec^2\left(\frac{\theta}{2}\right) \cdot \frac{1}{2} \cdot \frac{d\theta}{dt}$

    Given $\frac{d\theta}{dt} = k$, we have:

    $\frac{d}{dt}\left(\frac{dy}{dx}\right) = \frac{k}{2} \sec^2\left(\frac{\theta}{2}\right)$

    ---

    3. Evaluate at $\theta = \frac{\pi}{3}$

    Here $\frac{\theta}{2} = \frac{\pi}{6}$. We know $\sec(\pi/6) = \frac{2}{\sqrt{3}}$, so $\sec^2(\pi/6) = \frac{4}{3}$.

    Thus:

    $\frac{d}{dt}\left(\frac{dy}{dx}\right)\bigg|_{\theta=\pi/3} = \frac{k}{2} \cdot \frac{4}{3} = \frac{2k}{3}$

    ---

    Watch out

    A common mistake is to compute the slope itself at $\theta=\pi/3$ (which is $\tan(\pi/6)=1/\sqrt{3}$) and then multiply by $k$, forgetting the extra factor from the chain rule on $\tan(\theta/2)$. Always differentiate the slope expression fully.

    Tip

    The simplification $\frac{\sin\theta}{1+\cos\theta} = \tan\frac{\theta}{2}$ is a neat trigonometric shortcut that makes differentiation much cleaner.

    ---

    ✓Final answer

    The correct option is (D).

    ANSWER: D