Quiz – sections 2
Maximum & Minimum – Day 3 – “Area”
Example 8
A rectangular lot is bordered on one side by a stream and on the other three sides by 600 m of fencing. Find the dimensions of the lot if its area is a maximum.
Understand the problem
|Picture: |Given: |
| | |
|[pic] |We have 600 m of fencing. |
| |area = maximum |
| |Find: |
| | |
| |The dimensions of the lot. |
| | |
| |NOTE: [pic] |
Solution:
|We have 600 m of fencing. |Represent the two unknowns as one variable. |
| | |
|So,…………. |(We will represent them both as “w”.) |
|[pic] | |
|[pic] |We need to write our formula with one variable to represent the problem. |
|[pic] | |
|[pic] |Put the formula into standard form. |
|Now we are ready to COMPLETE THE SQUARE! |
| | |
|[pic] |
|So, the vertex is: (150, 45000) | |
|Now, translate the information to answer the question. |
| |
|The maximum area of the lot is 45000 m squared, but this wasn’t what we were asked to find. |
| |
|We see from the vertex, that the width of the lot is 150 m. We find the length using the formula we made at the start: |
|[pic] [pic] |
| |
|So, the dimensions of the lot are 150m by 300m. |
Example 9
A lifeguard marks off a rectangular swimming area at a beach with 200 m of rope. What is the greatest area of water that she can enclose?
Example 10
What is the maximum area of a triangle having 16 cm as the sum of its base and height?
Understand the problem
|Picture: |Given: |
| | |
| |16 = BASE + HEIGHT |
| |Find: |
| | |
| |The maximum area of a triangle. |
| | |
| |NOTE: [pic] |
Solution:
|16 = BASE + HEIGHT |Represent the two unknowns as one variable. |
|So,…………. | |
|[pic] |(We will represent them both as “h”.) |
|[pic] |We need to write our formula with one variable to represent the problem. |
|[pic] | |
|[pic] |Put the formula into standard form. |
|Now we are ready to COMPLETE THE SQUARE! |
| | |
|[pic] |
|So, the vertex is: (8, 32) | |
|Now, translate the information to answer the question. |
| |
|The maximum area of the triangle is 32 cm squared. |
| |
|We also know that the height of the triangle is 8 cm and we can find the base using the formula we created at the beginning: |
|[pic] [pic] |
Example 11
A rectangular area is to be enclosed by a fence. Two fences, parallel to one side of the field, divide the field into 3 equal rectangular fields. If 2400 m of fencing is available, find the dimensions of the field giving the maximum area.
Assignment:
1. Page 272-273 # 18, 19, 21 AND
2. A landscaper wishes to enclose a rectangular rest area with trees planted 1 m apart along the boundary. The perimeter of the area is to be 120 m. What will the maximum area of the rest area be?
-----------------------
width
length
STEP 1: Factor out the “a” value.
STEP 4: Rewrite the perfect square in its factored form and simplify if necessary.
STEP 3: Boot the negative out of the brackets.
STEP 2: Calculate and include the magic number!
BASE
HEIGHT
STEP 1: Factor out the “a” value.
STEP 3: Boot the negative out of the brackets.
STEP 4: Rewrite the perfect square in its factored form and simplify if necessary.
STEP 2: Calculate and include the magic number!
................
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