Suppose a spherical balloon is being filled with air so that its' volume and radius increase.

𝑡
=
time (in seconds)

𝑉
=
volume of the balloon at time
𝑡
(in cm
3
)

𝑟
=
radius of the balloon at time
𝑡
(in cm)

Restate the following sentence in terms of the above variables:

"How fast is the radius changing when the radius is 7 cm, if the balloon is being filled with air at a constant rate of 3 cm
3
/s?"

(Note: You do not need to solve the problem. Only determine what the question is asking, mathematically.)

Question

Suppose a spherical balloon is being filled with air so that its' volume and radius increase.

𝑡
=
  time    (in seconds)

𝑉
=
 volume of the balloon at time 
𝑡
     (in cm
3
)

𝑟
=
  radius of the balloon at time 
𝑡
      (in cm)

Restate the following sentence in terms of the above variables:

"How fast is the radius changing when the radius is 7 cm, if the balloon is being filled with air at a constant rate of 3 cm
3
/s?"

(Note: You do not need to solve the problem. Only determine what the question is asking, mathematically.)

BlackTom AI · Accepted Answer

The problem asks us to restate the given physical situation in mathematical terms using the variables V, r, and t, and to identify which option correctly expresses the requested derivative relationship at a specific radius.

Option 1: What is dr/dt when r = 7, given that dV/dt = 3 ?
- This matches the actual goal: determine the rate of change of the radius with respect to time at r = 7, given a constant rate of change of volume dV/dt = 3. Since V = (4/3)πr^3, differentiating both sides with respect to t yields dV/dt = 4πr^2 * dr/dt. Therefore, solving for dr/dt when r = 7 with dV/dt = 3 is exactly the asked quantity.

Option 2: What is dV/dt, if we know that r = 7 and dr/dt = 3
- This asks for the rate of change of volume given r and dr/dt, which is essentially dV/dt = 4πr^2 * dr/dt. While this is a valid relation, the stated question provides dV/dt and asks for dr/dt, not the reverse. It therefore addresses a different unknown.

Option 3: What is dV/d r if we know that r = 7 and V = 3
- dV/dr is the derivative of V with respect to r, which would be dV/dr = 4πr^2. However, the given data r = 7 and V = 3 does not directly yield dV/dr without additional context, and the question as posed concerns a time derivative dr/dt, not a spatial derivative. This option misaligns with the required relationship and the provided quantities.

Option 4: What is dr/dt when r = 7, if dV/dt = −3
- This mirrors Option 1 but changes the sign of dV/dt. Since dV/dt is negative, dr/dt would be negative as well (dr/dt = dV/dt / (4πr^2) = −3/(4π·49) ), so it describes a scenario where the balloon’s radius decreases. However, the core structure of the question remains the same; the negative sign simply yields a different numeric value, but the option still asks for the same type of derivative as Option 1, just with a different given rate.

Option 5: What is dV/dt, given that r = 7 and dV/dr = 3
- This asks for dV/dt from a given dV/dr, using a chain rule idea, but it mixes spatial and temporal derivatives improperly for the stated scenario. The correct chain would require dr/dt, not directly given, and the question is about dr/dt, not dV/dt, in this configuration. This option conflates derivatives and therefore does not reflect the asked quantity.

Overall, Option 1 directly aligns with the asked quantity and the established relationship between dV/dt and dr/dt derived from V = (4/3)πr^3, making it the appropriate choice for what the problem is requesting. The other options misinterpret the unknown, misstate the quantity to be found, or mix derivatives in a way that does not fit the target question.

MAT135H5_F25_ALL SECTIONS 4.1 Preparation Check

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