"Seemingly" Simple question about Area of Circle.

austinrman

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I am taking a free pre-algebra online course at Udacity. I, however, am stuck on a question. If you can not enter the correct answer on a practice quiz you can not move forward in the course, so I really need help with this one! Thanks for any help.

Here is the question:

You have a table in your living room with a round top that has a diameter of 2 and ½ feet. You would like to put a piece of glass over the top of the table to protect the surface. What is the Area of the tabletop? (Use 3.14 for the value of pi). Write your answer as a mixed number.


Here is my work and answer:

Area of tabletop = 3.14*radius squared
diameter= 2_1/2
radius = 1.25
so area= 3.14*1.25^2
this =4.91 rounded to nearest hudred
and 4.91 as a mixed number = 4_91/100.
Where did I go wrong?
 
You have a table in your living room with a round top that has a diameter of 2 and ½ feet. You would like to put a piece of glass over the top of the table to protect the surface. What is the Area of the tabletop? (Use 3.14 for the value of pi). Write your answer as a mixed number.


Here is my work and answer:

Area of tabletop = 3.14*radius squared
diameter= 2_1/2
radius = 1.25
so area= 3.14*1.25^2
this =4.91 rounded to nearest hundredth \(\displaystyle \ \ \ \) <----- Don't round here.
and 4.91 as a mixed number = 4_91/100.
Where did I go wrong?


Well, if the answer is desired to be a mixed number, then let's use fractions for all of the values at the start.

Area of tabletop = (157/50)*radius squared *
diameter = (5/2) feet
radius = (5/4) feet

*The pi approximation of 3.14 = 314/100 = 157/50

Area of tabletop = \(\displaystyle \bigg(\dfrac{157}{50}\bigg)\bigg(\dfrac{5}{4} \ feet\bigg)^2\)



\(\displaystyle = \ \bigg(\dfrac{157}{50}\bigg)\bigg(\dfrac{25}{16} \ feet^2\bigg)\)


\(\displaystyle = \ \bigg(\dfrac{157}{2}\bigg)\bigg(\dfrac{1}{16}\bigg)\ feet^2 \)


\(\displaystyle = \ \dfrac{157}{32} \ feet^2\)


\(\displaystyle = \ \dfrac{4(32) \ + \ 29}{32} \ feet^2 \)




\(\displaystyle = \ 4\frac{29}{32} \ feet^2\)
 
Note that 29/32= 0.90625 which, to two decimal places, would also be 0.91. There are many fractions that, rounded to two decimal places, would be 0.91. That is why, if you want your final answer as a fraction, it is better to do as lookagain did and work entirely in fractions, which are exact as opposed to rounded decimals that are only approximate.
 
Thanks so much. I am not sure why I never tried to do the work in all fractions. But now I see your point about not rounding because I could find the answer even working with the whole numbers with decimals:

3.14*1.25^2=4.90625

.90625 as fraction=90625/100,000

90525/100,000 simplified=29/32
 
Note that 29/32= 0.90625 which, to two decimal places, would also be 0.91. There are many fractions that, rounded to two decimal places, would be 0.91. That is why, if you want your final answer as a fraction, it is better to do as lookagain did and work entirely in fractions, which are exact as opposed to rounded decimals that are only approximate.

Where π is involved, you can never write an exact answer in numerals.
 
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