---
title: "Project 3: Cooling a Drink"
author: "Your Name"
format:
  html:
    toc: true
    code-fold: false
execute:
  echo: true
  warning: false
  message: false
---

## Model and assumptions

Explain the cooling-rate formula in your own words. Why does cooling slow down over time? State two assumptions.

## First step by hand

With a two-minute step, calculate the first Euler update. Then show the Euler prediction, predicted ending rate, average rate, and corrected RK2 update. Include units.

## Two-minute simulations

Complete the loop. Remove `#| eval: false` when your code is ready to run.

```{r}
#| eval: false

initial_temperature <- 90   # degrees C
ambient <- 22               # degrees C
k <- 0.15                   # per minute
target <- 47                # degrees C; owner's preference
end_time <- 12              # minutes

delta_t <- 2
time_2 <- seq(0, end_time, by = delta_t)
euler_2 <- numeric(length(time_2))
rk2_2 <- numeric(length(time_2))
euler_2[1] <- initial_temperature
rk2_2[1] <- initial_temperature

for (i in 2:length(time_2)) {
  # Euler: use the old Euler temperature to find its starting rate.
  # TODO: update euler_2[i].

  # RK2: use the old RK2 temperature to find its starting rate.
  # TODO: predict the endpoint using an Euler step.
  # TODO: find the rate at that prediction.
  # TODO: update rk2_2[i] using the average of the two rates.
}

results_2 <- data.frame(time = time_2, euler = euler_2, rk2 = rk2_2)
head(results_2, 2)
```

Compare the second row with your hand calculation. Do they agree?

## Graph and first recorded target time

Make a graph of the two-minute Euler and RK2 simulations and the exact cooling curve. Find the first recorded time at or below 47°C for each method. The pattern `which(temperature <= target)[1]` gives the position of the first qualifying value.

```{r}
#| eval: false

exact_2 <- ambient + (initial_temperature - ambient) *
  exp(-k * time_2)

# TODO: plot exact_2 versus time_2.
# TODO: add Euler and RK2 lines, a legend, and axis labels with units.

# TODO: use which() to find the first qualifying position for each method.
# TODO: look up the corresponding times in time_2.
```

Describe the difference between the three curves. Which simulated method first records a temperature at or below 47°C sooner?

## Half-minute simulations

Repeat both methods with a 0.5-minute step. Copy and adapt the loop above, using new vector names so that you keep the two-minute results. Remove `#| eval: false` when the code is ready.

```{r}
#| eval: false

delta_t <- 0.5
time_05 <- seq(0, end_time, by = delta_t)
euler_05 <- numeric(length(time_05))
rk2_05 <- numeric(length(time_05))
euler_05[1] <- initial_temperature
rk2_05[1] <- initial_temperature

for (i in 2:length(time_05)) {
  # TODO: Euler update using euler_05[i - 1].
  # TODO: RK2 prediction and corrected update using rk2_05[i - 1].
}

# TODO: find the first recorded time at or below 47 C for each method.
```

## Results and numerical error

Use the given exact formula to find the temperature at minute 12. Calculate the absolute error of all four simulated temperatures at minute 12. Create a small table containing the method, step size, minute-12 temperature, absolute error, and first recorded target time.

```{r}
#| eval: false

exact_at_12 <- ambient + (initial_temperature - ambient) *
  exp(-k * end_time)

# TODO: collect the four final temperatures and four first target times.
# TODO: calculate abs(simulated_temperature - exact_at_12).
# TODO: put the results in a data frame and display it.
```

How do method and step size affect the error? Remember that an accurate solution to a model is not proof that the model fits a real drink.

## Recommendation

How long should this owner wait according to your simulations? Explain your choice and one reason the prediction could differ from a real temperature measurement.