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Chapter 2: Problem 4

You have 16 miles of fence that you will use to enclose a rectangular field. a. Draw a picture to show that you can arrange the 16 miles of fence into a rectangle of width 3 miles and length 5 miles. What is the area of this rectangle? b. Draw a picture to show that you can arrange the 16 miles of fence into a rectangle of width 2 miles and length 6 miles. What is the area of this rectangle?

Short Answer

Expert verified
a. Area = 15 square miles; b. Area = 12 square miles.

Step by step solution

01

Understanding the Perimeter Constraint

For any rectangle, the perimeter is calculated as \( 2 imes ( \text{length} + \text{width} ) \). In this problem, the perimeter of the field, which is the length of the fence, is given as 16 miles. We'll use this to verify the dimensions in parts (a) and (b).
02

Arranging the Fence for Part (a)

For a rectangle with a width of 3 miles and a length of 5 miles, calculate the perimeter: \( 2 \times (5 + 3) = 2 \times 8 = 16 \) miles. This matches the given fence length, confirming that such a rectangle is possible. The area is calculated as \( \text{length} \times \text{width} = 5 \times 3 = 15 \) square miles.
03

Arranging the Fence for Part (b)

For a rectangle with a width of 2 miles and a length of 6 miles, calculate the perimeter: \( 2 \times (6 + 2) = 2 \times 8 = 16 \) miles. This also matches the given fence length, confirming that such a rectangle is achievable. The area is \( \text{length} \times \text{width} = 6 \times 2 = 12 \) square miles.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rectangles and Perimeters
A rectangle is a four-sided shape where opposite sides are equal in length and every angle is a right angle (90 degrees). When working with rectangles, understanding the concept of perimeter is crucial.
For a rectangle, the perimeter can be calculated using the formula:
  • Perimeter = 2 脳 (Length + Width)
This formula helps determine the total distance around the rectangle. In our original exercise, we used 16 miles of fencing, which means that the sum of all four sides of the rectangle must equal 16 miles.
This knowledge allows us to find different combinations of length and width that fit the requirement, and it鈥檚 important to understand that several rectangles with different dimensions can have the same perimeter.
Understanding Area Calculation in Rectangles
Area refers to the amount of space contained within a two-dimensional shape. For rectangles, the area is calculated using the formula:
  • Area = Length 脳 Width
In our exercise, by rearranging the dimensions of a rectangle while keeping the perimeter constant, we noticed how the area changes. For example:
  • A rectangle with dimensions 5 miles by 3 miles yields an area of \(5 \times 3 = 15\) square miles.
  • Another rectangle with dimensions 6 miles by 2 miles results in an area of \(6 \times 2 = 12\) square miles.
These variations demonstrate an important concept: while the perimeter might remain unchanged, the area can vary significantly based on the ratio of length to width. It's vital to understand this relationship, especially when optimizing layouts in real-world applications, such as allocating land.
Solving Algebraic Problems with Perimeter and Area
Algebraic problems involving perimeter and area can seem challenging at first, but breaking them down into smaller steps makes them manageable. In exercises like these, the initial step is usually to define the given values and determine the unknowns.
Let's say we are given a fixed perimeter, like in our 16-mile example, and we need to establish possible dimensions of the rectangle. This requires setting up an equation:
  • Given: Perimeter is 16 miles: \(2 \times (L+W) = 16\)
  • Solve for one variable, for instance, \(L\) or \(W\).
By manipulating these equations, we can explore the various possible combinations of dimensions that satisfy the perimeter condition. Additionally, we can further calculate and compare the areas to see how different dimensions affect the space enclosed by the rectangle.
Algebraic skills are essential here as they allow us to systemically solve real-world problems where balancing different dimensions for the best design is key.

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Most popular questions from this chapter

Two friends enjoy competing with each other to see who has the best time in running a mile. Initially (before they ever raced each other), the first friend runs a mile in 7 minutes, and for each race that they run, his time decreases by 13 seconds. Initially, the second friend runs a mile in 7 minutes and 20 seconds, and for each race that they run, his time decreases by 16 seconds. Which will be the first race in which the second friend beats the first?

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We are to buy quantities of two items: \(n_{1}\) units of item 1 and \(n_{2}\) units of item \(2 .\) a. If item 1 costs \(\$ 3.50\) per unit and item 2 costs \(\$ 2.80\) per unit, find a formula that gives the total \(\operatorname{cost} C\), in dollars, of the purchase. b. An isocost equation shows the relationship between the number of units of each of two items to be purchased when the total purchase price is predetermined. If the total purchase price is predetermined to be \(C=162.40\) dollars, find the isocost equation for items 1 and 2 from part a. c. Solve for \(n_{1}\) the isocost equation you found in part b. d. Using the equation from part c, determine how many units of item 1 are to be purchased if 18 units of item 2 are purchased.

You own a motel with 30 rooms and have a pricing structure that encourages rentals of rooms in groups. One room rents for \(\$ 85\), two rent for \(\$ 83\) each, and in general the group rate per room is found by taking \(\$ 2\) off the base of \(\$ 85\) for each extra room rented. a. How much money do you take in if a family rents two rooms? b. Use a formula to give the rate you charge for each room if you rent \(n\) rooms to an organization. c. Find a formula for a function \(R=R(n)\) that gives the revenue from renting \(n\) rooms to a convention host. d. What is the most money you can make from rental to a single group? How many rooms do you rent?

Motion toward or away from us distorts the pitch of sound, and it also distorts the wavelength of light. This phenomenon is known as the Doppler effect. In the case of light, the distortion is measurable only for objects moving at extremely high velocities. Motion of objects toward us produces a blue shift in the spectrum, whereas motion of objects away from us produces a red shift. Quantitatively, the red shift \(S\) is the change in wavelength divided by the unshifted wavelength, and thus red shift is a pure number that has no units associated with it. Cosmologists believe that the universe is expanding and that many stellar objects are moving away from Earth at radial velocities sufficient to produce a measurable red shift. Particularly notable among these are quasars, which have a number of important properties (some of which remain poorly understood). \({ }^{11}\) Quasars are moving rapidly away from us and thus produce a large red shift. The radial velocity \(V\) can be calculated from the red shift \(S\) using $$ V=c \times\left(\frac{(S+1)^{2}-1}{(S+1)^{2}+1}\right) $$ where \(c\) is the speed of light. a. Most known quasars have a red shift greater than 1. What would be the radial velocity of a quasar showing a red shift of 2 ? Report your answer as a multiple of the speed of light. b. Make a graph of the radial velocity (as a multiple of the speed of light) versus the red shift. Include values of the red shift from 0 to 5 . c. The quasar \(3 C 48\) shows a red shift of \(0.37\). How fast is \(3 C 48\) moving away from us? d. Find approximately the red shift that would indicate a radial velocity of half the speed of light. e. What is the maximum theoretical radial velocity that a quasar could achieve?
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