Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 3: Problem 47
In \(43-47,\) express each answer in simplest radical form. The dimensions of a rectangle are \(\sqrt{250}\) meters and \(\sqrt{1,440}\) meters. a. Express the perimeter of the rectangle in simplest radical form. b. Express the length of the diagonal of the rectangle in simplest radical form.
Short Answer
Expert verified
Perimeter: \( 34\sqrt{10} \) meters; Diagonal: \( 13\sqrt{10} \) meters.
Step by step solution
01
Understand the problem
We are given a rectangle with side lengths of \( \sqrt{250} \) meters and \( \sqrt{1440} \) meters. We need to find two things: (a) the perimeter and (b) the length of the diagonal in simplest radical form.
02
Calculate the Perimeter
The perimeter \( P \) of a rectangle is calculated using the formula \( P = 2(l + w) \), where \( l \) is the length and \( w \) is the width. Here, \( l = \sqrt{250} \) and \( w = \sqrt{1440} \). Therefore,\[ P = 2(\sqrt{250} + \sqrt{1440}). \] We need to simplify \( \sqrt{250} \) and \( \sqrt{1440} \) by finding their prime factors.
03
Simplify Each Square Root
To simplify \( \sqrt{250} \), note that 250 can be expressed as \( 25 \times 10 = 5^2 \times 2 \). Thus, \( \sqrt{250} = \sqrt{5^2 \times 10} = 5\sqrt{10} \). To simplify \( \sqrt{1440} \), note that 1440 can be expressed as \( 144 \times 10 = 12^2 \times 10 \). Thus, \( \sqrt{1440} = \sqrt{12^2 \times 10} = 12\sqrt{10} \).
04
Substitute Simplified Radicals into Perimeter Formula
Substitute the simplified forms into the formula for the perimeter: \[ P = 2(5\sqrt{10} + 12\sqrt{10}) = 2(17\sqrt{10}). \]This simplifies to: \[ P = 34\sqrt{10}. \]
05
Calculate the Diagonal Using the Pythagorean Theorem
The diagonal \( d \) of a rectangle is found using the Pythagorean theorem: \[ d^2 = l^2 + w^2. \] Hence, \[ d^2 = (5\sqrt{10})^2 + (12\sqrt{10})^2 = 250 + 1440. \]
06
Compute the Diagonal
Calculate \( d^2 = 250 + 1440 = 1690. \) The diagonal \( d \) is therefore \( \sqrt{1690} \). We need to simplify \( \sqrt{1690} \) further.
07
Simplify the Diagonal
The number 1690 can be expressed in terms of its prime factors: \( 1690 = 169 \times 10 = (13^2) \times 10 \). Thus, \[ \sqrt{1690} = \sqrt{(13^2) \times 10} = 13\sqrt{10}. \]
08
Conclusion
The perimeter of the rectangle in simplest radical form is \( 34\sqrt{10} \) meters, and the length of the diagonal is \( 13\sqrt{10} \) meters.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Perimeter of a Rectangle
When we talk about the perimeter of a rectangle, we're referring to the total distance around the rectangle. The formula for calculating the perimeter is simple:
- The formula is given by \( P = 2(l + w) \), where \( l \) is the length and \( w \) is the width of the rectangle.
- This formula effectively adds up all the sides of the rectangle. There are two lengths and two widths, which is why we multiply the sum by \( 2 \).
Diagonal of a Rectangle
In any rectangle, the diagonal is the line segment that connects opposite corners. This diagonal splits the rectangle into two right-angled triangles.To find the length of the diagonal, we often use the relationship from the properties of right triangles:
- The Pythagorean Theorem.
Pythagorean Theorem
The Pythagorean Theorem is a fundamental principle in geometry for right-angled triangles. It's expressed as \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse (the longest side of the triangle), and \( a \) and \( b \) are the other two sides. Here's how it works:
- In a rectangle, the diagonal acts as the hypotenuse of a right triangle.
- The two sides of the rectangle are the other two sides of the triangle.
- By applying the theorem, you can find the length of the diagonal.
Prime Factors
Prime factorization involves breaking down a number into the prime numbers that multiply together to make it. For instance, 250 can be split into prime factors as \( 5^2 \times 2 \). Simplifying radicals often involves prime factorization.Why prime factors?
- Prime factors help in simplifying complex square roots by identifying perfect squares within them.
- If you can find a perfect square within a number, it can be simplified outside the radical.
- For example, \( \sqrt{250} = \sqrt{5^2 \times 10} = 5\sqrt{10} \).
