A puppy gains weight, $w$ , at a rate approximately inversely proportional to its age, $t$ , in months. $\newline$ Which equation describes this relationship? $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{d w}{d t}=\frac{k}{w}$ $\newline$ (B) $\frac{d w}{d t}=k w$ $\newline$ (C) $\frac{d w}{d t}=k t$ $\newline$ (D) $\frac{d w}{d t}=\frac{k}{t}$

Full solution

Q. A puppy gains weight,

w

, at a rate approximately inversely proportional to its age,

t

, in months.

\newline

Which equation describes this relationship?

\newline

Choose

1

answer:

\newline

(A)

\frac{d w}{d t}=\frac{k}{w}

\newline

(B)

\frac{d w}{d t}=k w

\newline

(C)

\frac{d w}{d t}=k t

\newline

(D)

\frac{d w}{d t}=\frac{k}{t}

Define Inverse Proportionality: The problem states that the weight gain rate $\frac{dw}{dt}$ is inversely proportional to the puppy's age $t$ . This means that as the age increases, the rate of weight gain decreases, and vice versa. The concept of inverse proportionality can be expressed mathematically as one variable being equal to a constant divided by the other variable.
Use Constant of Proportionality: To represent this relationship with an equation, we use a constant of proportionality, which we can call $k$ . The equation that correctly represents an inverse proportionality between $\frac{dw}{dt}$ and $t$ is $\frac{dw}{dt} = \frac{k}{t}$ , where $k$ is a positive constant.
Match with Given Options: Now, we need to match this relationship with the given options. The option that states $\frac{dw}{dt} = \frac{k}{t}$ is the one that correctly represents an inverse proportionality between the rate of weight gain and the age of the puppy.