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2 months ago

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Two planes A and B can each fly through still air at 550 mph. The distance from Paris to Los Angeles is 5000 miles. Wind blows f

rom Los Angeles towards Paris at a speed of 100 mph. Plane A leaves Paris at noon flying towards Los Angeles. Plane B departs Los Angeles 1 hour later to fly to Paris. How far from Los Angeles are the two planes when they pass each other?

1 answer:

PIT_PIT [12.4K]2 months ago

5 0

Answer:2688.63 miles

Step-by-step explanation:

Given

The distance from Paris to Los Angeles $d=5000 miles$

Wind traveling at 100 mph from Los Angeles to Paris

The velocity of the plane through still air is $v=550 mph$

Relative velocity of Plane A is $v_a=550-100 =450 mph$ since Plane A is heading against the wind

Relative velocity of Plane B is $v_b=550+100=650 mph$

Plane A departs at noon

Distance covered by Plane A in 1 hour $=450\times 1=450 miles$

The total separation distance between the planes is $5000-450=4550 miles$

Assuming they meet after t hours

Distance covered by Plane B in that time frame $=v_b\times t=650\times t$

Distance traveled by Plane A in t hours is $=v_a\times t=450 t$

The total distance equals 4550 miles

$4550=650t+450t$

$t=\frac{4550}{1100}$

$t=4.13 hr$

Distance traveled by B is $=650\times 4.136=2688.63 miles$

thus, they meet at a point 2688.63 miles away from Los Angeles.

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The slope can be calculated using this formula:

$\hat \beta_1 =\frac{\sum (x-\bar x) (y-\bar y)}{\sum (x-\bar x )^2}$

Following the substitutions, we have:

$\hat \beta_1 =\frac{7625.9}{1248.9}=6.106$

The intercept can be determined with this formula:

$\hat \beta_o = \bar y -\hat \beta_1 \bar x$

Average values for x and y can be calculated this way:

$\bar X=7665.5/110 =69.686, \bar y= 17375/110=157.955$

Replacing yields:

$\hat \beta_o = 157.955 -6.106 (69.686)=-267.548$

Part b

The correlation coefficient can be calculated using the following formula:

$r=\frac{\sum (x-\bar x)(y-\bar y) }{\sqrt{[\sum (x-\bar x)^2][\sum(y-\bar y)^2]}}$

In our situation:

n=110, $\sum x_i y_i = \sum (X-\bar X)(Y-\bar Y) =7625.9,\sum x^2_i=\sum (x-\bar x)^2 =1248.9, sum y^2_i=\sum(y-\bar y)^2 =94228.8$

We can compute the correlation coefficient by substituting values:

$r=\frac{7625.9}{\sqrt{[1248.9][94228.8]}}=0.657$

The coefficient of determination is $r^2 = 0.657^2 =0.432$

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