# 4.3: Optimization

**At Grade**Created by: CK-12

*This activity is intended to supplement Calculus, Chapter 3, Lesson 7.*

## Problem 1 – Optimization of distance and area

Graph the line \begin{align*}y = 4x + 7\end{align*}. Find the point on the line that is closest to the origin.

- What point do you think minimizes the distance from the point to the origin?
- What function are you trying to minimize?
- What is the constraint?
- Write the function to minimize using one variable.

On the Home screen, find the exact coordinates that minimize the distance using the **Derivative** and **Solve** commands. To do this, find the first derivative, solve to find the critical value(s), and then find the second derivative to confirm a minimum.

- What are the \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*}coordinates of the point?
- What is the minimum distance?

Find the dimensions of a rectangle with perimeter 200 meters whose area is as large as possible.

- What dimensions do you think maximize the area?
- What function are you trying to maximize?

- What is the constraint?
- Write the function to maximize using one variable.

Find the dimensions that maximize the area using the **Derivative** and **Solve** commands.

- What are the dimensions of the rectangle?

## Problem 2 – Optimization of time derivative problems

A boat leaves a dock at 1 pm and travels north at a speed of 20 km/h. Another boat has been heading west at 15 km/h. It reaches the same dock at 2 pm. At what time were the boats closest together? Use \begin{align*}t\end{align*} for time.

- What is the position function for the boat heading north? West?

- What function are you trying to minimize?
- What is the constraint?
- Write the function to minimize using one variable.

Find the time at which the distance between the two boats is minimized using the **Derivative** and **Solve** commands.

- What is the minimum distance?
- What is the time at which this occurs? Remember to convert the value of \begin{align*}t\end{align*} to minutes.

## Extension – Parametric function

A projectile is fired with the following parametric functions:

\begin{align*}x = 500 \cos(30^\circ)t, y = 500 \sin(30^\circ)t - 4.9t^2\end{align*}

- What is the time when the projectile hits the ground?
- How far does it travel horizontally?
- What is the maximum height that it achieves?

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