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3 years ago

Find the distance between the pair of parallel lines.

Mathematics

2 answers:

Maurinko [17]3 years ago

6 0

Answer:

C) 3.58

Step-by-step explanation:

The distance between two parallel lines r1 and r2 is given by the absolute value of:

$d(r_1,r_2)=\frac{ax_1+by_1+c}{\sqrt{a^{2}+b^{2}}}$

Where a, b and c are the coefficients of the general form of the equation of the line r2 and x1 and y2 are the coordinates of one point that belongs to the line r1.

Let's call r1 the line define by: y = 2x +4 and r2 the line define by y=2x-4

So, the general form of the equation of line r2 is:

y = 2x - 4

0 = 2x - 4 - y

0 = 2x - y - 4

With this equation, a is equal to 2, b is equal to -1 and c is equal to -4.

Now, we need to found a point that belongs to line r1. This point can be found replacing x for any value and found y. If x is cero, for example, y is:

y = 2x +4

y = 2*0 + 4

y = 4

So, our point (x1, y1) is (0,4).

Replacing values of the equation of d(r1,r2), we get:

$d(r_1,r_2)=\frac{(2*0)+(-1*4)+(-4)}{\sqrt{2^{2}+(-1)^{2}}}=$

$d(r_1,r_2)=\frac{-8}{\sqrt{5}} = -3.58$

Then, the absolute value of -3.58 is 3.58, so the distance between the two parallel lines is 3.58.

guajiro [1.7K]3 years ago

6 0

Answer:

C) 3.58 is the correct answer.

Step-by-step explanation:

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3 years ago

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To solve for the missing steps, let's rewrite first the problem.

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Required:

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Answer:

Step-by-step explanation:

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3 years ago

A study by Consumer Reports showed that 64% of supermarket shoppers believe supermarket brands to be as good as national name br

Marianna [84]

Answer:

The calculated value |Z| = |-2.5| >2.326 at 0.05 level of significance

The alternative hypothesis is accepted at a 0.05 level of significance

The manager of Publix in Clemson believes 64% is too high for his own store

Step-by-step explanation:

Step:-1

Given that the consumer Reports showed that 64% of supermarket shoppers.

Given that the population proportion

P = 0.64

Given that random sample size 'n' = 100

Given that 52 believe the supermarket brands were as good as the national brands.

sample proportion

$p^{-} = \frac{x}{n} = \frac{52}{100} = 0.52$

Step:-2

Null hypothesis: The manager of the Publix in Clemson believes 64% is too low for his own store

μ < 0.64

Alternative Hypothesis:H₁:μ > 0.64

Test statistic

$Z = \frac{p-P}{\sqrt{\frac{PQ}{n} } }$

$Z = \frac{0.52-0.64}{\sqrt{\frac{0.64 X 0.36}{100} } }$

Z = -2.5

Level of significance = 0.05

Z₀.₀₅ = 2.326

The calculated value |Z| = |-2.5| >2.326 at 0.05 level of significance

Final answer:-

The null hypothesis is rejected at a 0.05 level of significance

The alternative hypothesis is accepted at a 0.05 level of significance

3 0

3 years ago

A. Find the amplitude.

Feliz [49]

Answers:

a) Amplitude = 2
b) Period = pi
c) Vertical shift = -2, which means it has been shifted down 2 units.
d) Horizontal shift = 3pi/8, this shifting is to the right.
e) There is  one  cycle between 0 and 2pi.
f) The equation of the graph is y = 2*sin(2(x-3pi/8))-2

========================================================

Explanations:

Part (a)

The highest point is when y = 0 and the lowest point is when y = -4. The vertical distance between the peak and valley is 4 units, which cuts in half to 2. This is the amplitude. It's the vertical distance from the center to either the peak or valley.

Note: Amplitude is always positive as it measures a distance.

---------------------

Part (b)

For x > 0, the first valley or lowest point occurs between 0 and pi/4. It appears to be the midpoint of the two values. So that would be (0+pi/4)/2 = pi/8.

The next valley occurs between pi and 5pi/4. Compute the midpoint to get (pi+5pi/4)/2 = (4pi/4+5pi/4)/2 = (9pi/4)/2 = 9pi/8

So we have the graph go from one valley x = pi/8 to the next valley over x = 9pi/8. This is a distance of pi units because 9pi/8-pi/8 = 8pi/8 = pi

The graph repeats itself every pi units, so the period is pi.

---------------------

Part (c)

The midline is normally through y = 0, aka the x axis. However, the graph shows the midline is through y = -2. This means the graph has been shifted down 2 units.

---------------------

Part (d)

This will depend on whether you use sine or cosine. This is entirely because cosine is a phase-shifted version of sine, and vice versa. I'll go with sine.

The parent sine function y = sin(x) goes through the origin (0,0)

However, as part (c) mentioned, we shifted the graph 2 units down. So we have y = sin(x)-2. But plugging x = 0 into this leads to the point (0,-2)

This doesn't match what the graph says. The graph shows the point (3pi/8, -2) on the red curve. The x coordinate 3pi/8 is the midpoint of pi/4 and pi/2

This must mean we need to shift the sine graph 3pi/8 units to the right.

---------------------

Part (e)

Start at the lowest point when x = pi/8. If you start the cycle here, then it ends when x = 9pi/8. See part (b).

So far we've completed one cycle. If we start at x = 9pi/8, then the next valley or lowest point is slightly beyond or to the right of x = 2pi. This means we run out of room and we haven't completed a full cycle.

Overall, one full cycle is between 0 and 2pi.

---------------------

Part (f)

Again I'm going to use sine instead of cosine. Refer back to part (d).

The general sine curve equation is

y = A*sin(B(x-C))+D

where

|A| = amplitude
B handles the period, specifically T = 2pi/B where T is the period. We can solve for B to get B = 2pi/T
C = horizontal phase shift
D = vertical shift, and ties together with the midline equation

In this case, we found that

|A| = 2
T = pi leads to B = 2pi/T = 2pi/pi = 2
C = 3pi/8
D = -2

So,

y = A*sin(B(x-C))+D

will update to

y = 2*sin(2(x-3pi/8))-2

which is one way to express the equation of the red curve. Optionally you can distribute the 2 through to (x-3pi/8).

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3 years ago