ASVAB circle and inscribed rectangle area problem

In summary, Rectangle ABCD is inscribed in the circle shown. If the length of side $\overline{AB}$ is 5 and the length of side $\overline{BC}$ is 12, then the area of the shaded region is $\frac{169\pi}{4}$.
  • #1
karush
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Rectangle ABCD is inscribed in the circle shown.
If the length of side $\overline{AB}$ is 5 and the length of side $\overline{BC}$ is 12
what is the area of the shaded region?

71.png


$a.\ 40.8\quad b.\ 53.1\quad c\ 72.7\quad d \ 78.5\quad e\ 81.7$

well to start with the common triangle of 12 5 13 gives us the diameter of 13
area of the circle is $\pi \left(\frac{13}{2}\right)^2=132.73$
area of {circle - retangle)=$132.73-60 =72.7$ which is c

typos maybe:unsure:
 
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  • #2
assuming $\overline{BC} = 12$, not $\overline{AB}$

$\pi \left(\dfrac{d}{2}\right)^2 - |AB| \cdot |BC|$
 
  • #3
I, because of my "prejudices", would immediately go to coordinate methods. Set up a Cartesian coordinate system with center at the center of the circle and with axes parallel to the sides of the rectangle.

The circle can be written $x^2+ y^2= R^2$. The line AB is x= -6 and CD is x= 6. The line BC is y= 5/2 and the line AD is y= -5/2.

The line y= 5/2 intersects the circle where $x^2+ 25/4= R^2$, so $x^2= R^2- 25/4$- that is at $\left(\sqrt{R^2- 25/4}, 5/2\right)$.
The line x= 6 intersects the circle where $36+ y^2= R^2$, so $y^2= R^2- 36$- that is at $\left(6, \sqrt{R^2- 36}\right)$.

Of course, those refer to the same point so we must have $6= \sqrt{R^2- 25/4}$ and $5/2= \sqrt{R^2- 36}$. The first is equivalent to $36= R^2- 25/4$ and the second to $25/4= R^2- 36$ so both give $R^2= 36+ 25/4= (144+ 25)/4= 169/4$ so $R= \pm\frac{13}{2}$. Since the radius of this circle is a positive number, $R= \frac{13}{2}$.

The circle has radius $\frac{13}{2}$ so area $\frac{169\pi}{4}$. The area of the rectangle is $12(5)= 60$ so the area of the shaded region is $\frac{169\pi}{4}- 60= \frac{169\pi- 240}{4}$.
 
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  • #4
well this would be a lot easier if it wasn't decimal
 
  • #5
skeeter said:
assuming $\overline{BC} = 12$, not $\overline{AB}$

$\pi \left(\dfrac{d}{2}\right)^2 - |AB| \cdot |BC|$
not sure if I have seen this notation before: |AB| as a distance between?
 
  • #6
karush said:
not sure if I have seen this notation before: |AB| as a distance between?

yes
 
  • #7
Jolly Good Mate!
 

1. What is the ASVAB circle and inscribed rectangle area problem?

The ASVAB (Armed Services Vocational Aptitude Battery) circle and inscribed rectangle area problem is a mathematical problem that involves finding the maximum area of a rectangle inscribed in a circle. This problem is commonly used in standardized tests, such as the ASVAB, to assess a person's ability to solve complex mathematical problems.

2. How do you solve the ASVAB circle and inscribed rectangle area problem?

To solve the ASVAB circle and inscribed rectangle area problem, you first need to understand the formula for finding the area of a rectangle, which is length x width. Then, you need to use the Pythagorean theorem to find the relationship between the length and width of the rectangle and the radius of the circle. Finally, you can use calculus to find the maximum area of the rectangle.

3. What is the Pythagorean theorem and how is it used in the ASVAB circle and inscribed rectangle area problem?

The Pythagorean theorem is a mathematical theorem that states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In the ASVAB circle and inscribed rectangle area problem, the Pythagorean theorem is used to find the relationship between the length and width of the rectangle and the radius of the circle.

4. What are some tips for solving the ASVAB circle and inscribed rectangle area problem?

Some tips for solving the ASVAB circle and inscribed rectangle area problem include understanding the formulas involved, carefully reading the problem and identifying the given information, drawing a diagram to visualize the problem, and using logical reasoning to come up with a solution. It is also important to double-check your calculations and make sure you have answered the question correctly.

5. How can solving the ASVAB circle and inscribed rectangle area problem benefit me?

Solving the ASVAB circle and inscribed rectangle area problem can benefit you by improving your critical thinking and problem-solving skills. It can also help you prepare for standardized tests, as this type of problem is commonly used in them. Additionally, understanding how to solve this problem can be useful in real-life situations, such as calculating the maximum area of a garden or determining the dimensions of a picture frame.

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