I will try and explain this since I can't draw the picture that went with this question.
A circular glass overlay was placed on the top of a circular table. Find the area of the shaded region, rounding to the nearest tenth.
The shaded region is the part of the table outside of the glass. the glass is in the center of the circle.
the information given is that that outside area ( the shaded part is 1 m).
The glass overlay has a radius of 1/2 m.
so I first found the area of the glass overlay. A=pi squared
pi*1/2 ^2 this is pi*1/4=.8
next I added the 1/2m(inner circle radius) and the 1m because together they give the radius of the full table.
using the same formula above for Area I got 7.1m
now I subtracted 7.1-.8=6.3m for the area of the shaded area.
If this was not too confusing to understand, does it seem like I did this correctly?
thank you for checking my work
You started well , but it collapsed on you in the second part.
I would not have rounded off intermediate answer, only round off your final answer.
I assume you used a calculator, get into the habit of using the memories of the calculator to store intermediate answers.
for area of glass overlay I got
A = .785398.. (I stored that )
area of table = 1.785398... (stored that)
let the radius of table be r
πr^2 = 1.785398..
r^2 = 1.785398/π
r = .7538..
I find a discrepancy in your question.
It said, "Find the area of the shaded region", but then you state that the shaded region has an area of 1 m^2
Anyway, we have all kinds of answers we can now play with
the shaded area(I mean portion, not the area as in the equation.) shows the distance of the shaded part=1m, . They put that there so you can add that piece onto 1/2m (radius of the inside circle) and together it gives you the radius of the full circle.
How did you get the radius of the table?
where did the 1.785398 come from?
I got the radius from adding the 1/2m from the inside circle and the 1m from the shaded in portion=1.5
put that in the equation getting 7.065
Can you please explain how you did you work? thank you
Correct me if I am wrong in the understanding of the question.
Round table, we don't know its radius, I eventually let it be r
Smaller circle lies on the table, the radius of the smaller table is 1/2 or .5 m
The area outside the circle is shaded and you said the area of that shaded region is 1 m^2
so I find the area of the circle to be
π(.5)^2 = .7853...
I added this to the shaded region to get the whole table area
= 1 + .7853 = 1.7853...
so now πr^2 = 1.7853
r^2 = 1.7853/π = .5686...
r = √.5686.. = .7539<---- radius of table
no, the 1 m is not the area of the shaded portion. there is a dotted line across the shaded part and ending at the outer part of the glass. the dotted part is 1 m. that is why I felt if you added that part to the radius of the glass, it would give the radius of the circular table.
Yes, it seems like you did the calculations correctly!
To find the area of the glass overlay, you correctly used the formula for the area of a circle, which is A = πr^2. Since the radius of the glass overlay is 1/2 m, you substituted that value into the formula and calculated A = π(1/2)^2 = 0.8 m^2.
Next, you found the combined radius of the glass overlay and the shaded area by adding the 1/2 m radius of the glass overlay to the 1 m radius of the shaded area. This gave you a total radius of 1/2 m + 1 m = 1.5 m.
Using the formula for the area of a circle once again, you found the area of the entire table by substituting the radius of 1.5 m into the formula. This gave you A = π(1.5)^2 = 7.1 m^2.
Finally, you subtracted the area of the glass overlay (0.8 m^2) from the area of the entire table (7.1 m^2) to calculate the area of the shaded region: 7.1 m^2 - 0.8 m^2 = 6.3 m^2.
Therefore, you correctly found that the area of the shaded region is approximately 6.3 m^2.