I don't know how to draw or do this.
A plane 5 cm from the center of a sphere intersects the sphere in a circle with diameter of 24 cm. Find the diameter of the sphere.
Draw a circle and draw a line cutting the circle, that line will represent the plane
So you are looking at a cross-section.
the length of the chord is 24 cm.
draw a line from the centre to that chord, it will be 5 cm.
Draw a line from the centre to the end of the chord, this would be the radius of the circle.
Don't you have a right-angled triangle where the hypotenuse is the radius??
so r^2 = 12^2 + 5^2
r = 13
so the diameter of the circle is 26
To find the diameter of the sphere, we can use the fact that the circle formed by the intersection of the plane and sphere has a diameter of 24 cm.
Let's break down the problem into steps:
Step 1: Find the radius of the circle.
Since the diameter of the circle is given as 24 cm, the radius can be found by dividing the diameter by 2:
Radius = Diameter / 2 = 24 cm / 2 = 12 cm.
Step 2: Determine the distance from the center of the sphere to the plane.
The problem states that the plane is 5 cm from the center of the sphere, so the distance from the center of the sphere to the plane is 5 cm.
Step 3: Find the radius of the sphere.
To find the radius of the sphere, we can use the Pythagorean theorem. The radius of the sphere, the radius of the circle, and the distance from the center to the plane form a right triangle. Let r be the radius of the sphere.
Using the Pythagorean theorem:
r^2 = (radius of the circle)^2 - (distance from the center to the plane)^2
r^2 = 12 cm^2 - 5 cm^2
r^2 = 144 cm^2 - 25 cm^2
r^2 = 119 cm^2
Therefore, the radius of the sphere is √119 cm.
Step 4: Calculate the diameter of the sphere.
The diameter of the sphere is twice the radius of the sphere:
Diameter = 2 * Radius = 2 * √119 cm.
Thus, the diameter of the sphere is 2√119 cm, which is approximately 2 * 10.92 cm = 21.84 cm (rounded to two decimal places).
To find the diameter of the sphere, we can use the Pythagorean Theorem and the properties of a circle.
Let's assume that the center of the sphere is point O, and the points where the plane intersects the sphere are points A and B. The line segment AB is the diameter of the circle with diameter 24 cm.
Since the plane is 5 cm from the center of the sphere, we can draw a line segment from the center of the circle to the midpoint of AB. Let's call this point M.
Now, we have a right triangle OMB, where OM is 5 cm (distance from the center of the sphere to the plane) and MB is half of the diameter of the circle, which is 12 cm.
Using the Pythagorean Theorem (a^2 + b^2 = c^2), we can solve for OB (the hypotenuse of the triangle):
OB^2 = OM^2 + MB^2
OB^2 = 5^2 + 12^2
OB^2 = 25 + 144
OB^2 = 169
Taking the square root of both sides:
OB = √169
OB = 13
Therefore, the diameter of the sphere is twice the length of OB, which is 2 * 13 cm = 26 cm.