Cross sections of the volume are washers or annuli with outer radii <em>x(y)</em> + 1, where
<em>y</em> = <em>x(y) </em>² - 1 ==> <em>x(y)</em> = √(<em>y</em> + 1)
and inner radii 1. The distance between the outermost edge of each shell to the axis of revolution is then 1 + √(<em>y</em> + 1), and the distance between the innermost edge of <em>R</em> on the <em>y</em>-axis to the axis of revolution is 1.
For each value of <em>y</em> in the interval [-1, 3], the corresponding cross section has an area of
<em>π</em> (1 + √(<em>y</em> + 1))² - <em>π</em> (1)² = <em>π</em> (2√(<em>y</em> + 1) + <em>y</em> + 1)
Then the volume of the solid is the integral of this area over [-1, 3]:
Answer:
0.4375
Step-by-step explanation:
Lets say that X is the random varaible that determine the arrival time of Bob and Y the random variable that determine the arrival time of Alice. Bot X and Y are Independent random variables with uniform [0,1] distribution. 15 minutes is the quarter of an hour, so we want to calculate P(|X-Y|) < 0.25.
Note that P(|X-Y| < 0.25) = P(|X-Y| < 0.25 | X ≥ Y) * P(X≥ Y) + P(|X-Y| < 0.25 | Y ≥ X) * P(Y ≥ X) = P(X-Y < 0.25 | X ≥Y) * P(X≥Y) + P(Y-X < 0.25 | Y≥X)*P(Y ≥ X).
For a simmetry argument, that expression is equivalent to 2*P(Y-X < 0.25 | Y≥X)*P(Y ≥ X) = 2*P(0 < Y-X < 0.25). The region 0 < Y-X < 0.25 is, for X between 0 and 0.75, a parallelogram, of base 0.75 and height 0.25, and for X between 0.75 and 1, it is a Triangle of base and height equal to 0.25. Therefore P(0 < Y-X < 0.25) = 0.25*0.75 + 0.25² * 0.5 = 7/32. Hence 2*P(0 < Y-X < 0.25) = 14/32 = 0.4375.
They will meet for lunch with probability 0.4375
Answer:
Therefore the measures of the angles are 22.5° and 67.5°.
Step-by-step explanation:
Complementary Angles:
Two angles are Complementary when they add up to 90 degrees.
Example 40° and 50° are Complementary Angles.
If 'x' and 'y' are Complementary Angles the we have
Here,
Let the common multiple be 'x '
Then the the Complementary Angles will be 'x' and '3x'.
....property of Complementary Angles.
Solving the equation we get
Substituting 'x' values we get
x= 22.5°
3x = 3 × 22.5 = 67.5°
Therefore the measures of the angles are 22.5° and 67.5°.
Answer:
The deer population decreased. The producers increased. So the number of different species increased.
Step-by-step explanation:
wolves are reintroduced ---> deer population decreases --> producer population begins to increase --> other species numbers increase as well