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# Geometry/Circles - Arc Length and Area

Question
Mr. Hussain,

Hello, I really hope you can help. It's pretty embarrassing, but my little sister asked for my help on her geometry homework. Unfortunately, I was never good in math. We found the answers online, but it is driving me crazy that I couldn't solve them. I have uploaded and attached the geometry problems.

Hello Caroline,

For Problem 1, we must find the sum of the circumferences of the circles, but only the portions before the intersection.  When an angle is formed at the center of the circle (such as the 80-degree and 100-degree angles), it subtends an arc of that same measure.  The circumference of a circle of radius r is given by 2πr, and we will have to multiply that by the fraction of the circumference desired.  This must done for both circles.

Therefore, the numerical answer is given by
2π6((360-80)/360)+2π4((360-100)/360), which rounds to 47 feet.

For Problem 2, we must first find the area of cake.  It is made up of a square of side 8 and a circle of radius 4 (the diameter is 8, so the radius is 8/2=4).  A square's area can be found by squaring the side length and a circle of radius r has area πr².

Therefore, the area of the cake is given by
8²+π4².

A quarter cup of icing covers 23 square inches, so we must divide the above area by 23 to determine how many quarter cups are required.  That division rounds to 5 quarter cups, so 1 and 1/4 cups are required.

Azeem
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Geometry

Volunteer

#### Azeem Hussain

##### Expertise

I welcome your questions on algebra, 2D and 3D geometry, parabolic functions and conic sections, and any other mathematical queries you may have.

##### Experience

4 years as a drop-in and by-appointment tutor at Champlain College. Private tutor for dozens of clients over the past 8 years.

Publications
CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry

Education/Credentials
Bachelor of Science, Major Mathematics and Major Economics, McGill University, 2014. Diploma of Collegiate Studies; Pure and Applied Science, Champlain College Saint-Lambert, 2010.