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A newborn calf weighs 40 kilograms. Each week its weight increases by
5%. Let
W be the weight in kilograms of the calf after
t weeks. Which of the following best explains the relationship between
t and
W ?
Choose 1 answer:
(A) The relationship is linear because
W increases by a factor of
5% each time
t increases by 1 .
(B) The relationship is exponential because
W increases by a factor of 1.05 each time
t increases by 1 .
(C) The relationship is exponential because
W increases by a factor of 5 each time
t increases by 1 .
(D) The relationship is linear because
W increases by 2 as
t increases from
t=0 to
t=1.

A newborn calf weighs $40$ kilograms. Each week its weight increases by $5 \%$ . Let $W$ be the weight in kilograms of the calf after $t$ weeks. Which of the following best explains the relationship between $t$ and $W$ ? $\newline$ Choose $1$ answer: $\newline$ (A) The relationship is linear because $W$ increases by a factor of $5 \%$ each time $t$ increases by $1$ . $\newline$ (B) The relationship is exponential because $W$ increases by a factor of $1$ . $05$ each time $t$ increases by $1$ . $\newline$ (C) The relationship is exponential because $W$ increases by a factor of $5$ each time $t$ increases by $1$ . $\newline$ (D) The relationship is linear because $W$ increases by $2$ as $t$ increases from $t=0$ to $t=1$ .

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Evaluate a linear function: word problems

Full solution

Q. A newborn calf weighs

40

kilograms. Each week its weight increases by

5 \%

. Let

W

be the weight in kilograms of the calf after

t

weeks. Which of the following best explains the relationship between

t

and

W

?

\newline

Choose

1

answer:

\newline

(A) The relationship is linear because

W

increases by a factor of

5 \%

each time

t

increases by

1

.

\newline

(B) The relationship is exponential because

W

increases by a factor of

1

.

05

each time

t

increases by

1

.

\newline

(C) The relationship is exponential because

W

increases by a factor of

5

each time

t

increases by

1

.

\newline

(D) The relationship is linear because

W

increases by

2

as

t

increases from

t=0

to

t=1

.

Initial Weight Calculation: The calf starts at $40$ kilograms and increases by $5\%$ each week. To find the weight after $1$ week, we calculate $40 + (5\% \text{ of } 40)$ . That's $40 + (0.05 \times 40) = 40 + 2 = 42$ kilograms.
Non-Linear Relationship Explanation: The increase is not by a constant amount but by a percentage, which means the relationship between $W$ and $t$ is not linear. It's a common mistake to think a percentage increase is linear.
Exponential Formula Derivation: Since the weight increases by a percentage each week, the formula for the weight after $t$ weeks is $W = 40 \times (1.05)^t$ . This is because each week the weight is multiplied by $1.05$ (which is $100\% + 5\%$ ).
Correct Relationship Identification: The correct relationship is exponential because the weight is multiplied by a factor of $1.05$ each time $t$ increases by $1$ . This matches option $(B)$ and not the other options.

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