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A newborn calf weighs $40\,\text{kilograms}$ . Each week its weight increases by $5\%$ . Let $W$ be the weight in kilograms of the calf after $n$ weeks. Which of the following best explains the relationship between $W$ and $n$ ?

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Find probabilities using the addition rule

Full solution

Q. A newborn calf weighs

40\,\text{kilograms}

. Each week its weight increases by

5\%

. Let

W

be the weight in kilograms of the calf after

n

weeks. Which of the following best explains the relationship between

W

and

n

?

Define variables: Let's define the variables: $\newline$ $W =$ weight of the calf after $t$ weeks $\newline$ $t =$ number of weeks since birth $\newline$ The initial weight of the calf is $40$ kilograms. $\newline$ Each week, the calf's weight increases by $5\%$ . $\newline$ To find the relationship between $W$ and $t$ , we need to use the formula for exponential growth, which is: $\newline$ $W =$ initial weight $*(1 +$ growth rate $)^{t}$ $\newline$ In this case, the initial weight is $40$ kilograms and the growth rate is $t$ $1$ in decimal form.
Calculate exponential growth: Now we will substitute the known values into the formula: $\newline$ $W = 40 \times (1 + 0.05)^t$ $\newline$ This simplifies to: $\newline$ $W = 40 \times (1.05)^t$ $\newline$ This equation shows that the weight of the calf after $t$ weeks is $40$ kilograms multiplied by $1.05$ raised to the power of $t$ .
Substitute known values: The relationship between $W$ and $t$ is exponential. As $t$ increases, $W$ increases at a rate of $5\%$ per week. This means that each week, the calf's weight is $105\%$ of the previous week's weight.

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