The surface area and the volume of a sphere are both 3 digit integers times π. If r is the radius of the sphere, how many integral values can be found for r?
Volume of sphere = (4/3)π r^3
You said the result is a 3 digit integer time π
So r^3 must be an integer if r is to be an integer
perfect cubes are 1,8,27, 64, 125, 216, 343, 512, 729, 1000
4/3 times those must remain a 3 digit integer.
which means the 3 must cancel, implying that the only ones to consider is
(4/3)(216) = 288 and (4/3)(729) = 972
So for the volume part of the question, we may have r = 6 or r = 9
the surface area is 4πr^2
if r = 6, then 4(6)^2 = 144 , ok!
ir r = 9, then 4(9)^2 = 324 , yeah, 3 digits
so r = 6 and r = 9 both satisfy your stated condition.
Well, you could call this problem "Spheres: The Search for Integral Values." It's like a mathematical adventure! Let's dive in and solve it.
The formula for the surface area of a sphere is A = 4πr^2, and the formula for the volume is V = (4/3)πr^3.
We are given that both A and V are 3-digit integers times π. Let's start by looking at A.
To have a 3-digit integer times π, our equation would be 4πr^2 ≈ 100π.
Dividing both sides by 4π, we get r^2 ≈ 25.
Simplifying further, we can say that r ≈ √25, which means r can be approximately ±5.
Now let's examine V.
For V to be a 3-digit integer times π, our equation would be (4/3)πr^3 ≈ 100π.
Dividing both sides by (4/3)π, we get r^3 ≈ 75.
We can simplify this to r ≈ ∛75, which is approximately 4.85 (rounded to two decimal places).
Since we are looking for integral values for r, we need to round this down to 4.
So, we have two possible integral values for r: -5 and 4.
Therefore, we can find two integral values for r in this case.
To solve this problem, we need to find integral values for the radius of the sphere, given that both the surface area and volume are three-digit integers times π.
Let's start by considering the formula for the surface area of a sphere:
Surface Area = 4πr²
And the formula for the volume of a sphere:
Volume = (4/3)πr³
Since both the surface area and volume are three-digit integers times π, we can rewrite these equations as:
Surface Area = Xπ
Volume = Yπ
Where X and Y are three-digit integers.
Substituting the equations for surface area and volume, we get:
4πr² = Xπ
(4/3)πr³ = Yπ
Simplifying and canceling out π:
4r² = X
(4/3)r³ = Y
Now, we can find possible integral values for r by considering the possible values for X and Y.
Since X is a three-digit integer, its possible values range from 100 to 999.
Substituting these values into the equation 4r² = X, we can solve for r:
100 ≤ 4r² ≤ 999
25 ≤ r² ≤ 249.75
Taking the square root of both sides, we get:
5 ≤ r ≤ 15.8
Since r must be an integral value, we round down the upper limit to 15.
Therefore, the possible integral values for r are from 5 to 15, inclusive. Hence, there are 11 integral values for r.
To determine the integral values that can be found for the radius (r) of the sphere, we need to analyze the information given regarding the surface area and volume.
Let's start with the formula for the surface area (A) of a sphere:
A = 4πr²
Since we are informed that the surface area is a 3-digit integer times π, we can express it as:
A = kπ, where k is a 3-digit integer.
By substituting this value into the surface area formula and rearranging the equation, we have:
4πr² = kπ
Dividing both sides of the equation by π:
4r² = k
Now let's move on to the formula for the volume (V) of a sphere:
V = (4/3)πr³
Again, we are told that the volume is a 3-digit integer times π, so we can express it as:
V = mπ, where m is a 3-digit integer.
By substituting this value into the volume formula and rearranging, we have:
(4/3)πr³ = mπ
Dividing both sides of the equation by π and multiplying by 3:
4r³ = 3m
Now we have a system of equations:
4r² = k
4r³ = 3m
To find the integral values of r, we can iterate through possible values of k and m, and calculate the corresponding values for r. We need to check if both equations are satisfied to determine if a particular value of r is valid.
Let's analyze the possible values of k and m:
Since both k and m are 3-digit integers, they can vary from 100 to 999.
For each value of k between 100 and 999, we calculate the corresponding value of r from the first equation by evaluating:
r = √(k/4)
Similarly, for each value of m between 100 and 999, we calculate the corresponding value of r from the second equation by evaluating:
r = ∛(3m/4)
We then check if the value of r obtained from both equations is an integer. If it is, then it is a valid integral value for r.
By iterating through all possible values of k and m, counting the number of valid integral values for r will give us the final answer.
Note that we can optimize this process by acknowledging that r must be an integer if both 4r² and 4r³ are integers. So, we only need to iterate through possible values of k, calculate r using the first equation, and check if both 4r² and 4r³ are integers.
Performing this process will give us the number of integral values that can be found for r.