The volume of a sphere can be calculated from its radius as follows:

Therefore, squaring both sides, we get

Substituting:

If we let the constant of variation be , we see that

,

and  varies directly as , the sixth power of .

This may seem confusing, but is pretty straightforward.

Thus, for every 125 meters below the surface, the temperature decreases by one degree.

To find how much it decreases with every 100 meters, we need to do the following:

Thus, the temperature decreases by  every 100 meters.

When two quantities vary inversely, their products are always equal to a constant, which we can call k. If the square of x and the cube of y vary inversely, this means that the product of the square of x and the cube of y will equal k. We can represent the square of x as x² and the cube of y as y³. Now, we can write the equation for inverse variation.

x²y³ = k

We are told that when x = 8, y = 8. We can substitute these values into our equation for inverse variation and then solve for k.

8²(8³) = k

k = 8²(8³)

Because this will probably be a large number, it might help just to keep it in exponent form. Let's apply the property of exponents which says that a^ba^c = a^b+c.

k = 8²(8³) = 8²⁺³ = 8⁵.

Next, we must find the value of y when x = 1. Let's use our equation for inverse variation equation, substituting 8⁵ in for k.

x²y³ = 8⁵

(1)²y³ = 8⁵

y³ = 8⁵

In order to solve this, we will have to take a cube root. Thus, it will help to rewrite 8 as the cube of 2, or 2³.

y³ = (2³)⁵

We can now apply the property of exponents that states that (a^b)^c = a^bc.

y³ = 2^3•5 = 2¹⁵ 

In order to get y by itself, we will have the raise each side of the equation to the 1/3 power.

(y³)^(1/3) = (2¹⁵)^(1/3)

Once again, let's apply the property (a^b)^c = a^bc.

y^{(3 • 1/3)} = 2^{(15 • 1/3)}

y = 2⁵ = 32

The answer is 32. 

varies directly as and inversely as , so for some constant of variation ,

.

We can square both sides to obtain:

.

, so .

By substitution,

.

Using as the constant of variation, we see that varies inversely as the square of and  directly as the fourth power of .

The volume of a cylinder can be calculated from its height and the radius of its base using the formula:

, so ;

, so .

The volume is 1,000, and by substitution, using the other equations:

If we take as the constant of variation, we get

,

meaning that varies inversely as .

varies directly as the square of and the cube root of , and inversely as the fourth root of , so, for some constant of variation ,

We take the reciprocal of both sides, then extract the square root:

Taking as the constant of variation, we see that varies directly as the square root of and the eighth root of , and inversely as the sixth root of .

The formula for inverse variation is as follows: 

Use the x and y values from the first part of the sentence to find k. 

Then use that k value and the given x value to find y.

To find the degree of a polynomial we must find the largest exponent in the function.

The degree of the polynomial  is 5, as the largest exponent of  is 5 in the second term. 

When a polynomial has more than one variable, we need to find the degree by adding the exponents of each variable in each term. 

 has a degree of 4 (since both exponents add up to 4), so the polynomial has a degree of 4 as this term has the highest degree. 

When a polynomial has more than one variable, we need to find the degree by adding the exponents of each variable in each term.

Even though  has a degree of 5, it is not the highest degree in the polynomial - 

 has a degree of 6 (with exponents 1, 2, and 3). Therefore, the degree of the polynomial is 6.