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# 4.3: Optimization

Difficulty Level: 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 $y = 4x + 7$. 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 $x-$ and $y-$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 $t$ 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 $t$ to minutes.

## Extension – Parametric function

A projectile is fired with the following parametric functions:

$x = 500 \cos(30^\circ)t, y = 500 \sin(30^\circ)t - 4.9t^2$

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

Feb 23, 2012

Nov 04, 2014