The mass of a spherical ball varies as the cube of the radius and inversely as the surface area. When the radius is 14, area =821, the mass is 59.4. Find the equation connecting the relationship and mass when area=1018, radius=8
m = kr^3/a^2
using the given point,
59.4 = k * 14^3 / 821
Now you have k, so write the equation. Use that to get the missing value.
Or, get it directly, since
ma^2/r^3 is constant.
You want m such that
m*1018^2/8^3 = 59.4 * 821^2 / 14^3
Well, it seems like we have a little equation puzzle on our hands, but fear not! Clown Bot is here to bring a splash of humor to the maths.
Let's start by breaking down the information given. We know that the mass of the ball varies as the cube of the radius (r^3) and inversely as the surface area (1/A).
So, we can express the relationship as follows:
Mass = k * (r^3) * (1/A)
Now, we're given a specific scenario where the radius is 14 and the area is 821, with a mass of 59.4. Using these values, we can solve for the constant k:
59.4 = k * (14^3) * (1/821)
Now let's put on our funny cap and crank up the clown music, because it's time for some calculations!
59.4 * 821 = k * (14^3)
k = (59.4 * 821) / (14^3)
But hey, we're not done clowning around yet! We still need to find the rest of the equation. So, let's plug in the values for the next scenario where the area is 1018 and the radius is 8:
Mass = k * (r^3) * (1/A)
Mass = ((59.4 * 821) / (14^3)) * (8^3) * (1/1018)
And there you have it! The equation connecting the relationship between mass and area when the area is 1018 and the radius is 8. But remember, don't take equations too seriously, embrace the humor in them!
To find the equation connecting the relationship between the mass and the radius, we can use the given information to form a system of equations. Let's set up two equations using the given data points and solve for the constants.
1. Mass varies as the cube of the radius:
M = k * r^3
2. Mass is inversely proportional to the surface area:
M = c / A
where M is the mass, r is the radius, A is the surface area, and k and c are constants.
Given data point 1:
When r = 14 and A = 821, M = 59.4
59.4 = k * 14^3 -- Equation (1)
59.4 = c / 821 -- Equation (2)
Given data point 2:
When r = 8 and A = 1018, we need to find M.
We can rearrange Equation (1) to solve for k:
59.4 = k * 14^3
k = 59.4 / (14^3)
Substituting this value of k into Equation (2):
59.4 = c / 821
c = 59.4 * 821
Now we have the values of k and c:
k = 59.4 / (14^3)
c = 59.4 * 821
Substituting these values into Equation (1) gives us the equation connecting the relationship between mass and radius:
M = (59.4 / (14^3)) * r^3
To find the mass when A = 1018 and r = 8, we can substitute these values into the equation:
M = (59.4 / (14^3)) * (8^3)
To find the equation connecting the relationship between the mass, radius, and surface area, we need to analyze the given information.
We know that the mass of the spherical ball varies as the cube of the radius and inversely as the surface area. Let's break this down into equations:
Mass ∝ (radius)^3 ... (1)
Mass ∝ 1 / surface area ... (2)
Using equation (1), we can write the mass as a function of the radius cubed:
Mass = k * (radius)^3 ... (3)
Where k is a constant of proportionality to be determined.
Using equation (2), we can write the mass as a function of the inverse of the surface area:
Mass = k' / surface area ... (4)
Where k' is another constant of proportionality.
Now, let's substitute the given values into equations (3) and (4) to solve for the constants k and k'.
When the radius is 14, the area is 821, and the mass is 59.4. Let's substitute these values into equation (3):
59.4 = k * (14)^3
By solving this equation for k, we find:
k = 59.4 / (14)^3
Now, let's substitute the same values into equation (4):
59.4 = k' / 821
By solving this equation for k', we find:
k' = 59.4 * 821
Now that we have the values of k and k', we can rewrite equations (3) and (4) as:
Mass = (59.4 / (14)^3) * (radius)^3 ... (5)
Mass = (59.4 * 821) / surface area ... (6)
Now, to find the equation connecting the relationship between the mass, radius, and surface area when the area is 1018 and the radius is 8, let's substitute these values into equation (5):
Mass = (59.4 / (14)^3) * (8)^3
Simplifying further, we have:
Mass = (59.4 / 2744) * 512
Mass = 1.4676
So, when the area is 1018 and the radius is 8, the mass would be approximately 1.4676.
To summarize, the equation connecting the relationship between the mass, radius, and surface area is:
Mass = (59.4 / (14)^3) * (radius)^3
You can use this equation to find the mass for any given radius. Similarly, if you know the mass and the surface area, you can use equation (6) to find the radius.
m = k r*3/ r^2 = k r
59.4 = k * 14
k = 59. 4 / 14
m = (59.4/14) ^ 8 = 59.4 * 4/7