Let's analyze each scenario to determine whether the relationship can be modeled by a linear, exponential, or neither type of function.

(a) Total cost of producing $n$ calculators:

• The problem states there is a fixed cost of $2000 to develop the calculator, and it costs $20 to manufacture each one. This means the total cost $C(n)$ can be expressed as: $C(n) = 2000 + 20n$ This is a linear equation because it follows the form $C(n) = m n + b$, where $m$ (slope) is 20, and $b$ (y-intercept) is 2000.

Answer: Linear

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(b) Population of a bacteria colony after $t$ weeks:

• The colony starts with 300 organisms and doubles every week. The population $P(t)$ at time $t$ can be expressed as: $P(t) = 300 \times 2^t$ This is an exponential function because the population grows by a constant factor (doubling) over equal time intervals.

Answer: Exponential

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(c) College cost growing by 6% per year after 1998:

• The cost of attending college grows by a constant percentage (6% per year). This type of growth is modeled by an exponential function: $C(t) = 10,069 \times (1.06)^t$ This is an exponential equation because the cost increases by a constant percentage, not by a fixed amount.

Answer: Exponential

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(d) Area of a triangle in terms of its base $b$:

• The height $h$ is equal to the base $b$, and the formula for the area $A$ of a triangle is: $A = \frac{1}{2} \times \text{base} \times \text{height}$ Since the height equals the base, this becomes: $A = \frac{1}{2} b^2$ This is neither linear nor exponential, as the relationship involves a quadratic term ($b^2$).

Answer: Neither

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Summary:

• (a) Linear
• (b) Exponential
• (c) Exponential
• (d) Neither

Would you like more details on any of the explanations, or have any other questions?

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Here are 5 follow-up questions to deepen understanding:

1. How can you distinguish between linear and exponential growth in real-life scenarios?
2. What kind of function would represent the cost of manufacturing calculators if there were bulk discounts?
3. What happens to the bacteria population if the doubling rate changes every week?
4. Can you think of another example where costs increase by a percentage every year?
5. How would the area formula change if the height was not equal to the base in the triangle?

Tip: Linear functions show constant change (e.g., fixed cost per item), while exponential functions show percentage-based growth or decay (e.g., doubling time or growth rate).