1. The surface area of a cube varies directly as the square of its edge. Find its surface are.

2. A certain object traveling at 24 meters per second has a kinetic energy of 72 joules. If the kinetic energy of that object varies directly as the square of its velocity, find the kinetic energyof that object travelling at 27 m/s.

Our topic here in direct square variation

1. To find the surface area of a cube, we need to determine the formula for direct square variation. Let's denote the surface area as A and the edge length as e.

According to the problem, the surface area varies directly as the square of the edge length. Mathematically, this can be written as:

A ∝ e^2

To remove the proportionality symbol (∝), we need to introduce a constant of variation, which we'll represent as k. Therefore, the equation becomes:

A = k * e^2

2. In this case, we have an object traveling at 24 m/s with a kinetic energy of 72 J. We want to find the kinetic energy of the same object traveling at 27 m/s.

Let's denote the kinetic energy as K and the velocity as v.

According to the problem, the kinetic energy varies directly as the square of the velocity. This can be expressed as:

K ∝ v^2

To remove the proportionality symbol (∝), we'll introduce a constant of variation, which we'll represent as k. So, the equation becomes:

K = k * v^2

To solve direct square variation problems, we use the formula:

y = kx^2

where y represents the dependent variable, x represents the independent variable, and k is the constant of variation.

Let's solve the given problems step by step:

1. The surface area of a cube varies directly as the square of its edge. Find its surface area.
In this case, the surface area is the dependent variable (y) and the edge length is the independent variable (x). We are given a direct square variation, so we can write the equation as:

y = kx^2

To find the constant of variation (k), we need some additional information. Let's assume the surface area is S and the edge length is e. We know that when the edge length is 1 unit, the surface area is also 1 unit^2. So we can write:

1 = k(1^2)

1 = k

Now, we can substitute the value of k back into the equation:

S = 1e^2

S = e^2

Therefore, the surface area of a cube is equal to the square of its edge length.

2. A certain object traveling at 24 meters per second has a kinetic energy of 72 joules. If the kinetic energy of that object varies directly as the square of its velocity, find the kinetic energy of that object traveling at 27 m/s.

In this case, the kinetic energy is the dependent variable (y) and the velocity is the independent variable (x). We are given a direct square variation, so we can write the equation as:

y = kx^2

To find the constant of variation (k), we need some additional information. Let's assume the kinetic energy is K and the velocity is v. Given that K = 72 when v = 24:

72 = k(24^2)

72 = k(576)

k = 72/576

k = 1/8

Now we can substitute the value of k and the new velocity into the equation:

K = (1/8)(27^2)

K = (1/8)(729)

K = 91.125 joules

Therefore, the kinetic energy of the object traveling at 27 m/s is 91.125 joules.

a = ks^2

E = kv^2
72 = k(24^2)
Now use that k to find

E = k(27^2)

I got it now. Thanks much Steve.