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3 years ago

8

Because of the increase in traffic between Springfield and Orangeville, a new road was built to connect the two towns. The old r

oad goes south x miles from Springfield to Freeport and then goes east x + 5 miles from Freeport to Orangeville. The new road is 9 miles long and goes straight from Springfield to Orangeville. Find the number of miles that a person drove using the old road. Please help ASAP!!! :(

1 answer:

Doss [256]3 years ago

5 0

<h3>Answer:</h3>

11.70 miles

<h3>Explanation:</h3>

The road from Springfield to Orangeville is apparently the hypotenuse of a right triangle whose legs are x and x+5. The value of x can be found from the Pythagorean theorem:

... 9² = x² + (x+5)²

... 2x² +10x -56 = 0 . . . . . . eliminate parentheses, collect terms, subtract 81.

... x² +5x -28 = 0 . . . . . . . . divide by 2 (because we can)

... x² +5x +6.25 = 28 +6.25 . . . . add 28, then add (5/2)² to complete the square

... (x +5/2) = √34.25 . . . . . . . .take the positive square root

... x = -2.5 +√34.25 ≈ 3.352

The total distance via the old road is the sum of the leg lengths:

... x + (x+5) = 2x +5 = 2(3.352) +5 = 11.704 . . . . miles

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As the complete question is here, however some of the values are not readable, thus the question is found online and is attached herewith.

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So the value of E(x) is 3.6667.

Part 8

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$E(x,y)=\int\limits^{5}_1 \int\limits^{10}_2 {xyf_{x,y}(x,y)} \,dy\, dx\\E(x,y)=\int\limits^{5}_1 \int\limits^{10}_2 {xy\frac{1}{16}} \,dy\, dx\\E(x,y)=\int\limits^{5}_1 \frac{x}{16}\left[\frac{y^2}{2}\right]^{10}_2\, dx\\E(x,y)=\int\limits^{5}_1 3x\, dx\\\\E(x,y)=3\left[\frac{x^2}{2}\right]^5_1\\E(x,y)=36$

So the value of E(x,y) is 36

Part 9

The value is given as

$Cov(X,Y)=E(x,y)-E(x)E(y)\\Cov(X,Y)=36-(3.6667)(4.6667)\\Cov(X,Y)=18.8886\\$

So the value of Cov(x,y) is 18.8886

Part 10

The variables X and Y are considered independent when

$f_{xy}(x,y)=f_x(x).f_y(y)\\$

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$f_x(x).f_y(y)=\frac{x-1}{8}\frac{10-y}{32} \\$

And

$f_{xy}(x,y)=\frac{1}{16}$

As these two values are not equal, this indicates that X and Y are not independent variables.

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3 years ago