Question

directions: for each of the following scenarios, identify whether the relationship described is linear, quadratic, exponential, or none of these. then give a reason for your conclusion. 1. your local library holds a penny drive in which people in the community are asked to donate their spare pennies. when you stop by, you hear someone announce that they will donate 1 penny today, 2 pennies tomorrow, 4 pennies the next day, and will continue to double the number of pennies that she donates each day after, for 2 weeks. 2. kieran has an air - conditioning unit installed in one window of his house. droplets of water fall at regular intervals from the air conditioner, creating a circular puddle on his patio. kieran tries to calculate the puddles area, and observes that the puddles radius increases by 1 centimeter every minute. 3. jevon is on vacation in new york city and wants to use a ride - sharing service. one particular ride - sharing service charges a flat fee of $2.50 and an additional charge of $0.35 per mile. 4. chenglei tracks the temperature for a particular day in september. she notices that it was 54°f at 6 a.m., 60°f at 9 a.m., 67°f at 12 p.m., 60°f at 3 p.m., and 54°f at 6 p.m.

Explanation:

Step1: Analyze Scenario 1 pattern

The number of pennies doubles daily: $1, 2, 4, 8,...$ This follows $y = 2^{x-1}$, where $x$ = day number, $y$ = pennies.

Step2: Analyze Scenario 2 relationship

Radius $r(t) = t$ (cm, $t$=minutes). Area $A(r) = \pi r^2 = \pi t^2$, a quadratic form.

Step3: Analyze Scenario 3 cost structure

Total cost $C(m) = 2.50 + 0.35m$, linear form $y=mx+b$.

Step4: Analyze Scenario 4 temperature trend

Temperatures rise to a peak then fall symmetrically, matching $y = ax^2 + bx + c$ shape.

Answer:

1. Exponential: The number of donated pennies is multiplied by a constant factor (2) each day, which follows an exponential growth pattern.
2. Quadratic: The area of the circular puddle is $\pi r^2$, and since the radius increases linearly with time, substituting gives a quadratic relationship between area and time.
3. Linear: The total ride cost follows the linear formula $C = 0.35m + 2.50$, where cost increases at a constant rate per mile.
4. Quadratic: The temperature values form a symmetric, U-inverted curve (rises to a midday peak then falls), which matches the shape of a quadratic function.