Question

If the radii of two intersecting spheres are 12 cm and 9 cm and the distance between their center is 15 cm, find the area of largest circle

Answer 1

Oh that's an interesting problem. I have some ideas, what level math are you in?

Answer 2

It might be helpful to sketch this situation.
Just supposing you move the spheres so that they just touch one another in one spot. Their centers would be (12 cm + 9 cm) apart. There's no circle!
But imagine that we push the smaller sphere into the other so that the centers of the two spheres are 15 cm (instead of 21 cm) apart. The intersection of the two spheres is a circle. All points on this circle are common to both the larger and the smaller sphere.

It appears you'll need to determine the radius of this circle of intersection. Once you have that, you can calculate the area defined by this circle.

Answer 3

@hitaro9 solid mensuration

Answer 4

@mathmale can you pls help me provide a sketch? I'm really confused :(

Answer 5

Can you pls explain more help me pls :( @mathmale

Answer 6

Hint:


Look at the image attachment as a reference


Imagine you are looking straight down at the spheres. So you are seeing a 2D cross section.


Step 1) Pick any sphere and place it at the origin. Let's say we pick the one with radius 12. Since we're looking at cross sections, we're placing the circle with radius 12 at the origin. This circle has the equation `x^2 + y^2 = 144`. Point A is the center of this circle.


Step 2) Plot a point at (0,12). This is point C


Step 3) Draw a horizontal line that is 9 units long that has (0,12) as an endpoint


Step 4) Draw another circle centered at (-9,12) with radius 9. This is the cross section of the other sphere. Point D is the center of this other circle.


Step 5) Plot points for the intersection of the two circles. One of these two points is C. The other point, we'll call it point E


Step 6) Find the midpoint of point C and point E. Call this point F


Step 7) Find the distance from F to C, or the distance from F to E. This distance will be the radius of the circular intersected region between the two spheres.

Answer 7

So how can i find the area using pir^2 with that radius?

Answer 8

Help pls

Answer 9

You use that radius in the formula \[A=\pi r^2\]to find the area of the circle made by the intersection of the two spheres (which is at right angles to your screen as you view the diagram Jim provided).

Answer 10

Imagine that you have a ball in your left hand, and a ball in your right hand. Only the ball in your right hand is made up of a magical material that can pass between the atoms of the ball in your left hand. As you squish the two of them together, the smaller ball in your right hand partially merges with the ball in your left hand. If you could stop at that point and slice through at the exact point where the two balls meet, you would have a circle, and the radius of that circle would be the distance between F and C on that diagram. You can find the distance between two points whose coordinates are \((x_1,y_1)\) and \((x_2,y_2)\) with the formula
\[d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\]which a bit of reflection should tell you is simply the Pythagorean theorem in a modest disguise. Now you know the radius of your intersection circle. Plug it into \[A = \pi r^2\]and you are done.

It occurs to me that perhaps you are having difficulty determining the coordinates of the two points E and F. Is that the case? I'm afraid I have to leave for the evening, so I can't help you through that right now, but maybe someone else can...I'll check back tomorrow when I get some time.

Answer 11

Have they given you an answer?

Answer 12

Not yet

Answer 13

Distance between the centers=15
radius of one sphere=12
radius of other sphere=9
Draw a perpendicular from the point of intersection on the line joining the two centers.
It is the radius of the required circle.
Let distance of foot of perpendicular from one center=x
Then distance of this point from other center=15-x
then apply Pythagoras's therem

Answer 14

whoa. what are you doing? you've given it away!