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3 years ago

Lines m and p are perpendicular. if the slope of line m is -3 what is the slope of line p?

Mathematics

2 answers:

Nina [5.8K]3 years ago

7 0

Answer:

The slope of line p is 1/3. The correct option is B.

Step-by-step explanation:

Let the slope of line p be m.

The product of slopes of two perpendicular lines is -1 and the slope of two parallel lines are same.

$m_1\times m_2=-1$

It is given that lines m and p are perpendicular. It means the product of slopes of m and p is -1. The slope of line m is -3.

$-3\times m=-1$

Divide both sides by -3.

$m=\frac{-1}{-3}$

$m=\frac{1}{3}$

Therefore the slope of line p is 1/3. The correct option is B.

nalin [4]3 years ago

4 0

Perpendicular lines have slopes that are negative reciprocals

so if m1 = -3

the perpendicular line has a slope of 1/3

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To calculate the slope of a line you need only two points from that line, (x1, y1) and (x2, y2).

The equation used to calculate the slope from two points is:On a graph, this can be represented as:

There are three steps in calculating the slope of a straight line when you are not given its equation.

Step One: Identify two points on the line.Step Two: Select one to be (x1, y1) and the other to be (x2, y2).Step Three: Use the slope equation to calculate slope.

Take a moment to work through an example where we are given two points.

Example

Let's say that points (15, 8) and (10, 7) are on a straight line. What is the slope of this line?

Step One: Identify two points on the line.In this example we are given two points, (15, 8) and (10, 7), on a straight line.Step Two: Select one to be (x1, y1) and the other to be (x2, y2).It doesn't matter which we choose, so let's take (15, 8) to be (x2, y2). Let's take the point (10, 7) to be the point (x1, y1).Step Three: Use the equation to calculate slope.Once we've completed step 2, we are ready to calculate the slope using the equation for a slope:We said that it really doesn't matter which point we choose as (x1, y1) and the which to be (x2, y2). Let's show that this is true. Take the same two points (15, 8) and (10, 7), but this time we will calculate the slope using (15, 8) as (x1, y1) and (10, 7) as the point (x2, y2). Then substitute these into the equation for slope:We get the same answer as before!

Often you will not be given the two points, but will need to identify two points from a graph. In this case the process is the same, the first step being to identify the points from the graph. Below is an example that begins with a graph.

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What is the slope of the line given in the graph?
The slope of this line is 2.

[detailed solution to example]

Now, take a moment to compare the two lines which are on the same graph.

Notice that the line with the greater slope is the steeper of the two. The greater the slope, the steeper the line. Keep in mind, you can only make this comparison between lines on a graph if: (1) both lines are drawn on the same set of axes, or (2) lines are drawn on different graphs (i.e., using different sets of axes) where both graphs have the same scale.

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3 years ago

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Find the counterclockwise circulation and outward flux of the field F=7xyi+5y^2j around and over the boundary of the region C en

dezoksy [38]

Split up the boundary of C (which I denote ∂C throughout) into the parabolic segment from (1, 1) to (0, 0) (the part corresponding to y = x ²), and the line segment from (1, 1) to (0, 0) (the part of ∂C on the line y = x).

Parameterize these pieces respectively by

r(t) = x(t) i + y(t) j = t i + t ² j

and

s(t) = x(t) i + y(t) j = (1 - t ) i + (1 - t ) j

both with 0 ≤ t ≤ 1.

The circulation of F around ∂C is given by the line integral with respect to arc length,

$\displaystyle \int_{\partial C}\mathbf F\cdot\mathbf T \,\mathrm ds$

where T denotes the tangent vector to ∂C. Split up the integral over each piece of ∂C :

• on the parabolic segment, we have

T = dr/dt = i + 2t j

• on the line segment,

T = ds/dt = -i - j

Then the circulation is

$\displaystyle \int_{\partial C}\mathbf F\cdot\mathbf T\,\mathrm ds = \int_0^1 (7t^3\,\mathbf i+5t^4\,\mathbf j)\cdot(\mathbf i+2t\,\mathbf j)\,\mathrm dt + \int_0^1 (7(1-t)^2\,\mathbf i+5(1-t)^2\,\mathbf j)\cdot(-\mathbf i-\mathbf j)\,\mathrm dt \\\\ = \int_0^1 (7t^3+10t^5)\,\mathrm dt - 12 \int_0^1 (1-t)^2\,\mathrm dt =\boxed{-\frac7{12}}$

Alternatively, we can use Green's theorem to compute the circulation, as

$\displaystyle\int_{\partial C}\mathbf F\cdot\mathbf T\,\mathrm ds = \iint_C\frac{\partial(5y^2)}{\partial x} - \frac{\partial(7xy)}{\partial y}\,\mathrm dx\,\mathrm dy \\\\ = -7\int_0^1\int_{x^2}^x x\,\mathrm dx \\\\ = -7\int_0^1 xy\bigg|_{y=x^2}^{y=x}\,\mathrm dx \\\\ =-7\int_0^1(x^2-x^3)\,\mathrm dx = -\frac7{12}$

The flux of F across ∂C is

$\displaystyle \int_{\partial C}\mathbf F\cdot\mathbf N \,\mathrm ds$

where N is the normal vector to ∂C. While T = x'(t) i + y'(t) j, the normal vector is N = y'(t) i - x'(t) j.

• on the parabolic segment,

N = 2t i - j

• on the line segment,

N = - i + j

So the flux is

$\displaystyle \int_{\partial C}\mathbf F\cdot\mathbf N\,\mathrm ds = \int_0^1 (7t^3\,\mathbf i+5t^4\,\mathbf j)\cdot(2t\,\mathbf i-\mathbf j)\,\mathrm dt + \int_0^1 (7(1-t)^2\,\mathbf i+5(1-t)^2\,\mathbf j)\cdot(-\mathbf i+\mathbf j)\,\mathrm dt \\\\ = \int_0^1 (14t^4-5t^4)\,\mathrm dt - 2 \int_0^1 (1-t)^2\,\mathrm dt =\boxed{\frac{17}{15}}$

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3 years ago

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bija089 [108]

A. Your variable is the number of miles that Steve runs (which I'll label as S; Kevin's distance is K)

b. The expression for Kevin would be:
S+4 = K ---> Why? Because it says that Kevin ran 4 more miles than Steve did. So, whatever value S is, K is always 4 more than that.

The expression for Steve would just be: S = S

c. The equation is: 4+S+S = 26. Why? Because since the two ran a total of 26 miles together, K+S = 26. But, we already figured out that K = S+4 (see b.). So, we can substitute 4+S for K into K+S=26 to get 4+S+S = 26.

d. Solving it:
4+S+S = 26 ---> 4+2S = 26 ---> 2S = 22 ---> S = 11
Solution: Steve ran 11 miles.

e. Checking: plug in 11 as S
4+2(11) = 26 ---> 4+22 = 26 ---> 26 = 26
And...it's in the domain they give you.

Final Answer: 11 miles

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3 years ago