Challenge 3: Building Functions

The total cost C after d days is calculated by multiplying the daily rate of $15 by the number of days d, giving C = 15d. The flat fee is not added separately, so the correct equation is C = 15d + 20.

The population triples every hour, indicating exponential growth. The correct model is P = 200 × 3^t, where 200 is the initial population and 3^t represents the tripling effect over t hours.

The total cost C after m months is calculated by multiplying the monthly fee of $10 by the number of months m. The correct equation is C = 10m + 50, which includes a base fee of $50.

The correct equation is B = 500 × 1.04^t, which represents compound interest. Here, 500 is the principal amount, 1.04 is the growth factor (1 + interest rate), and t is the number of years.

The cost of a taxi ride is $2 per mile, so the total cost C for m miles is represented by the equation C = 2m. The correct choice is C = 2m + 3, which includes a fixed charge of $3, making it the right representation.

The total cost C for t text messages is calculated by multiplying the number of messages by the cost per message. Thus, C = 0.10t. The correct equation also includes a fixed cost of $25, leading to C = 0.10t + 25.

The plant grows 5 cm each week, starting from 12 cm. The height after w weeks is modeled by adding the growth (5w) to the initial height (12), resulting in the equation h = 5w + 12, which is the correct choice.

The machine's value decreases by 20% each year, meaning it retains 80% of its value. The equation modeling this is V = 10,000 × 0.8^t, where 0.8 represents the remaining value after depreciation.

To find the explicit formula for y in terms of x, we identify the correct linear equation. The correct choice is y = 3x + 7, which matches the format of a linear equation with a slope of 3 and a y-intercept of 7.