Matt is standing on top of a cliff 305 feet above a lake. the measurement of the angle of depression to a boat on the lake is 42. how far is the boat from matt

≈ 410.4427 feet (this is rounded -- it's an infinite decimal)

Step-by-step explanation:

If you draw it out, you get a right triangle. If Matt is standing 305 feet above a lake, that is the altitude of the triangle. The angle of depression means, for example, the angle looking down from something at something else, in this case, from the cliff to the boat. The 42° is the acute angle between the altitude and the hypotenuse, the one at the top of the triangle. You want to find how far away the boat is from Matt directly, so that would be the hypotenuse, just connecting a straight line from him to the boat.

To answer these problems there are three tools you use to find the Sine, Cosine, and Tangent. Let's see which one would best help us in this situation.

The definition of sine is the side opposite the acute angle over the hypotenuse.

The definition of cosine is the side adjacent to the acute angle over the hypotenuse.

The definition of tangent is the side opposite the acute angle over the side adjacent to it.

In this case, you know you have the side adjacent to the acute angle, over the hypotenuse. Because you have the acute angle (42°) and the side adjacent to it (305) and the hypotenuse (which is what you're trying to find the value of. Which definition would fit this figure? Cosine, because it's side adjacent over hypotenuse.

Once you have that straightened out, you know that you need to find the cosine of 42°, which, if you look in a sine, cosine, and tangent chart (or type 42 into your calculator and hit cos) is 0.7431. Now you can write the equation .

Working with that equation, you need to get x by itself to figure out its value. How you do that is by multiplying both sides by x. This would look like . When you multiply 305 by x, you are multiplying by the number you were just dividing by so that answer is 305. I'll give you an example with different numbers: 6/2 then 6*2 is just 6.

What your equation looks like now is . How do you find x? Well it's the same as doing this: . Then you would divide 8 by 2 to get x, which equals 4. So, going back to our task at hand, . That's your answer, which is how far Matt is from the boat. Depending on which digit they want you to round the answer to you could punch in 305/0.7431 into a calculator and see many more digits and a much longer number. But don't forget to include units in your answer -- feet.

answer: learning objective(s)

·         describe the creation and use of systems of equations.

·         graph a system of linear equations on the coordinate plane and identify its solution.

introduction

sometimes graphing a single linear equation is all it takes to solve a mathematical problem. other times, one line just doesn’t do it, and a second equation is needed to model the situation. this is often the case when a problem involves two variables. solving these kinds of problems requires working with a system of equations, which is a set of two or more equations containing the same unknowns.

let’s take a look at systems of equations, and see what the graphs of individual equations within a system reveal about the mathematical relationship of the variables.

systems of equations

a system of equations contains two or more linear equations that share two or more unknowns. to find a solution for a system of equations, we must find a value (or range of values) that is true for all equations in the system.

the graphs of equations within a system can tell us how many solutions exist for that system. look at the images below. each show two lines that make up a system of equations (in the graph on the right the two lines are superimposed and look like a single line). how many points in common does each system of lines reveal?

one solution

no solutions

infinite solutions

if the graphs of the equations intersect, then there is one solution that is true for both equations.

if the graphs of the equations do not intersect (for example, if they are parallel), then there are no solutions that are true for both equations.

if the graphs of the equations are the same, then there are an infinite number of solutions that are true for both equations.

remember, the graph of a line represents every point that is a possible solution for the equation of that line. so when the graphs of two equations cross, the point of intersection lies on both lines, meaning that it is a possible solution for both equations. when the graphs of two equations never touch, there are no shared points and there are no possible solutions for the system. when the graphs of two equations lie on top of one another, they share all their points and every one is a possible solution.

graphing as a solution method

graphing equations in order to identify a specific point of intersection is usually not a precise way to solve systems because it is often difficult to see exactly where two lines intersect (unless you are using a computer-based graphing program that allows you to zoom in on a point). however, the graph of a system of equations can still give a good idea of what type of solution, if any, exists. let’s graph a system, and see how it works.

graph the system y = 3x and x + 2y = 4. how many solutions does this system have?

a graph of the two lines y = 3x and x + 2y = 4 shows us that the lines intersect, meaning that there is a single (x, y) value that satisfies both equations. looking at the graph does not tell us exactly where that point is, but we don’t need to know that information here, because the question only asks for the number of solutions.

so a system made of two intersecting lines has one solution. now let’s look at a different situation:

how many solutions exist for the system y -0.5x = 7 and 2y = x − 3?

plotting both equations, it looks like there is no solution—the lines are parallel. to check this finding, we can compare the slopes of the equations. to make that easy, we’ll rearrange the equations into slope-intercept form, or y = mx + b.   that gives us the equations y = 0.5x + 7 and y = 0.5x − 1.5.   yes, the slope of both lines is 0.5, which means the lines really are parallel. they never intersect, so there is no point that lies on both lines, and no solution to the system.

step-by-step explanation:

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step-by-step explanation: