First calculate the slope of the line using the slope formula.

Substituting in the known information.

Now use point slope form to find the equation of the line passing through these points.

Now identify which point lies on the line.

Therefore, the point that lies on the line is (9,10)

Find the least common denominator to solve this problem

Multiply 27 with , and multiply  with 3 to obtain common denominators.

Convert the fractions.

Combine the terms as one fraction.

The answer is:  

Simplify the numerator

Pull an x out of each term in the numerator

The x in the numerator and the x in the denominator cancel

Given

First FOIL the first equation. FOIL means to multiply each term in the first binomial with each term in the second binomial.

Now solve the second equation for .

Substitute the equation for  into the first equation to get a new equation only in terms of .

Distribute to eliminate the parentheses and simplify.

Now use the quadratic formula.

Given a quadratic in the form, 

For this particular question,

From here recall that if the value under the radical sign equals zero than  results in having just one solution.

Therefore, set the value that is under the radical equal to zero and solve for .

Since the binomials containing  are the same, set one equal to zero and solve.

This question is testing one's ability to analyze a function algebraically and recognize different but equivalent forms. Identifying properties of functions through analyzing equivalent forms is critical to this concept. Such properties that can be found through analyzing the different forms of a function include finding roots (zeros), extreme values, symmetry, and intercepts.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Recognize the general form of the function.

This is known as the difference of squares.

Step 2: Identify what is known.

Step 3: Substitute the known values into the difference of perfect squares found in step 1.

Step 4: Answer the question.

To find the positive solution set each binomial equal to zero and solve for x.

Therefore the positive solution is four.

To calculate the width of the crater, use the given information to establish that the image draws similar triangles. When triangles that have corresponding angles and a ratio to their side lengths they are considered to be similar triangles.

Identify the known information.

therefore,

and the bases of the triangles are parallel.

Also,

Set up the side ratios for this particular problem.

Looking at the only full ratio that is given, the scalar multiplier can be found.

Therefore, to find the width of the crater  multiply  by two.

First, solve this equation for y and then substitute the answer into the second equation:

Now substitute into the second equation and solve for x:

To solve for x, add the coefficients on the x variables together then divide both sides by three. 

To solve for  first identify what is known.

The question states that  is a right triangle whose  and   . It is important to recall that any triangle has a sum of interior angles that equals 180 degrees.

Therefore, to calculate  use the complimentary angles identity of trigonometric functions.

and since , then

Given the two equations, substitute the numerical value of  into the second equation to solve for . 

Substituting the numerical value for  into the equation with  is as follows.

From here, distribute the three.

Now square both side of the equation.

Remember to square both terms within the parentheses. Also, recall that squaring a square root sign cancels them out.

To solve for  divide both sides by six.