Given f '(x) = (2 - x)(6 - x), determine the intervals on which f(x) is increasing or decreasing. (2 points)Decreasing (-∞, 2); increasing on (6, ∞)
Decreasing (2, 6); increasing on (-∞, 2) U (6, ∞)
Decreasing (-∞, 2) U (6, ∞); increasing on (2, 6)
Increasing (-∞, -2) U (-6, ∞); increasing on (-2, -6)

Answers

Answer 1
Answer:

Answer:

Decreasing (2, 6); increasing on (-∞, 2) U (6, ∞)

Step-by-step explanation:

To determine where f(x) is increasing or decreasing, we set f'(x)=0 and check for the intervals.

We see that f'(x)=0 when either x=2 or x=6. Therefore, we'll need to check the intervals (-\infty,2), (2,6), and (6,\infty)

For the interval (\infty,2), we can pick x=0 . This means thatf'(0)=(2-0)(6-0)=(2)(6)=12>0, showing that f(x) increases on the interval (-\infty,2)

For the interval (2,6), we can pick x=4. This means that f'(4)=(2-4)(6-4)=(-2)(2)=-4<0, showing that f(x) decreases on the interval (2,6)

For the interval (6,\infty), we can pick x=7. This means that f'(7)=(2-7)(6-7)=(-5)(-1)=5>0, showing that f(x) increases on the interval (6,\infty)

Therefore, f(x) is increasing on (-\infty,2)\cup(6,\infty) and is decreasing on (2,6).


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The half-life of a radioactive substance is 200 years. There are 8000 grams of the substance initially. How many grams of the substance are left after 300 years?

Answers

Answer:

There are 2,000 grams left after 300 years.

Step-by-step explanation:

Giving the following information:

The half-life of a radioactive substance is 200 years. There are 8000 grams of the substance initially.

First, we need to calculate the reduction of the substance each year:

Yearly reduction= 8,000/400= 20 grams per year

Now, for 300 years:

300 year reduction= 20*300= 6,000

There are 2,000 grams left after 300 years.

1. 2x-y≤-6
2. 5x+4y ≥20

Answers

Answer

2x-y <or =-6

2x<or=-6+y

divide both sides by 2

x<or=1/2y+3

5(1/2y+3)+4y>or=20

5/2y+15+4y>or=20

5/2y+4y>or=20-15

13/2y>or=5

divide both sides by 2/13

y>or=10/13

2x-10/13<or=-6

2x<or=-6+10/13

2x<or=-68/13

divide both sides by 2

x< or =-34/13

Driscoll Middle School held basketball tryouts after school. Out of the 80 students, only 25% made the team. How many students made it on the team?

Answers

20 students made the basketball team

You and a group of friends are going to a five-day outdoor music festival during spring break. You hope it does not rain during the festival, but the weather forecast says there is a 15% chance of rain on the first day, a 10% chance of rain on the second day, a 20% chance of rain on the third day, a 20% chance of rain on the fourth day, and a 60% chance of rain on the fifth day. Assume these probabilities are independent of whether it rained on the previous day or not. What is the probability that it does not rain during the entire festival

Answers

Answer:

Probability that it does not rain during the entire festival = 0.19584

Step-by-step explanation:

We are given that you and a group of friends are going to a five-day outdoor music festival during spring break.

Also, Probability that there may be rain on first day, P(A) = 0.15

Probability that there may be rain on second day, P(B) = 0.10

Probability that there may be rain on third day, P(C) = 0.20

Probability that there may be rain on fourth day, P(D) = 0.20

Probability that there may be rain on fifth day, P(E) = 0.60

It is also provided that these probabilities are independent of whether it rained on the previous day or not.

Now, probability that it does not rain during the entire festival = Probability that there may not be rain on all five days

= (1 - P(A)) * (1 - P(B)) * (1 - P(C)) * (1 - P(D)) * (1 - P(E))

= (1 - 0.15) * (1 - 0.10) * (1 - 0.20) * (1 - 0.20) * (1 - 0.60)

= 0.85 * 0.90 * 0.80 * 0.80 * 0.40 = 0.19584

As a quality control manager for her distributing company, Zoe has 875.23 pounds of apples where she found 143.875 pounds of apples were rotten. How many pounds of fresh apples does she have? show your work.

Answers

Answer:

731.355

Step-by-step explanation:

875.230 - 143.875 = 731.355

apples       rotten       fresh

4. Find the volume of the given solid bounded by the elliptic paraboloid z = 4 - x^2 - 4y^2, the cylinder x^2 + y^2 = 1 and the plane z = 0.5. Sketch the region of integration and change the order of integration.

Answers

Answer:

2.5π units^3

Step-by-step explanation:

Solution:-

- We will evaluate the solid formed by a function defined as an elliptical paraboloid as follows:-

                                  z = 4 - x^2 -4y^2

- To sketch the elliptical paraboloid we need to know the two things first is the intersection point on the z-axis and the orientation of the paraboloid ( upward / downward cup ).

- To determine the intersection point on the z-axis. We will substitute the following x = y = 0 into the given function. We get:

                                 z = 4 - 0 -4*0 = 4

- The intersection point of surface is z = 4. To determine the orientation of the paraboloid we see the linear term in the equation. The independent coordinates ( x^2 and y^2 ) are non-linear while ( z ) is linear. Hence, the paraboloid is directed along the z-axis.

- To determine the cup upward or downwards we will look at the signs of both non-linear terms ( x^2 and y^2 ). Both non-linear terms are accompanied by the negative sign ( - ). Hence, the surface is cup downwards. The sketch is shown in the attachment.

- Theboundary conditions are expressed in the form of a cylinder and a plane expressed as:

                                x^2 + y^2 = 1\n\nz = 4

- To cylinder is basically an extension of the circle that lies in the ( x - y ) plane out to the missing coordinate direction. Hence, the circle ( x^2 + y^2 = 1 ) of radius = 1 unit is extended along the z - axis ( coordinate missing in the equation ).

- The cylinder bounds the paraboloid in the x-y plane and the plane z = 0 and the intersection coordinate z = 4 of the paraboloid bounds the required solid in the z-direction. ( See the complete sketch in the attachment )

- To determine the volume of solid defined by the elliptical paraboloid bounded by a cylinder and plane we will employ the use of tripple integrals.

- We will first integrate the solid in 3-dimension along the z-direction. With limits: ( z = 0 , z = 4 - x^2 -4y^2 ). Then we will integrate the projection of the solid on the x-y plane bounded by a circle ( cylinder ) along the y-direction. With limits: ( y = - √(1 - x^2) , y =  √(1 - x^2) ). Finally evaluate along the x-direction represented by a 1-dimensional line with end points ( -1 , 1 ).

- We set up our integral as follows:

                            V_s = \int\int\int {} \, dz.dy.dx

- Integrate with respect to ( dz ) with limits: ( z = 0 , z = 4 - x^2 -4y^2 ):

                           V_s = \int\int [ {4 - x^2 - 4y^2} ] \, dy.dx

- Integrate with respect to ( dy ) with limits: ( y = - √(1 - x^2) , y =  √(1 - x^2) )

                        V_s = \int [ {4y - x^2.y - (4)/(3) y^3} ] \, | .dx\n\nV_s = \int [ {8√(( 1 - x^2 ))  - 2x^2*√(( 1 - x^2 )) - (8)/(3) ( 1 - x^2 )^(3)/(2) } ] . dx

- Integrate with respect to ( dx ) with limits: ( -1 , 1 )

                       V_s =  [ 4. ( arcsin ( x ) + x√(1 - x^2) ) - (arcsin ( x ) - 2x ( 1 -x^2 )^(3)/(2) + x√(1 - x^2)  )/(2)  - ( 3*arcsin ( x ) + 2x ( 1 -x^2 )^(3)/(2) + 3x√(1 - x^2)  )/(3) ] | \limits^1_-_1\n\nV_s =  [ (5)/(2) *arcsin ( x ) + (5)/(3)*x ( 1 -x^2 )^(3)/(2) + (5)/(2) *x√(1 - x^2) ) ] | \limits^1_-_1\n\nV_s =  [ (5\pi )/(2)  + 0 + 0 ] \n\nV_s = (5\pi )/(2)

Answer: The volume of the solid bounded by the curves is ( 5π/2 ) units^3.

Final answer:

The volume of the bounded region is found by setting up a triple integral, changing to cylindrical coordinates, and integrating to get 3.5π. The region of integration is a solid capped by an elliptic paraboloid, lying inside the unit circle above the xy-plane. Changing the order of integration doesn't apply here as the given order is already the most ideal.

Explanation:

The subject of this question is

Calculating Volume

in integral calculus, specifically dealing with triple integrals. Given the equations z = 4 - x^2 - 4y^2, x^2 + y^2 = 1, and z = 0, we find the volume by setting up a triple integral. In cylindrical coordinates, this is  ∫ ∫ (4 - x^2 - 4y^2) rdrdθ from θ=0 to 2π and r=0 to 1. Changing to cylindrical coordinates, x = rcosθ and y = rsinθ, gives  ∫ ∫ (4 - r^2) rdrdθ. This evaluates to π(4r - (r^2)/2) evaluated from 0 to 1, which simplifies to π(4 - 0.5) = 3.5π.

Sketching the Region of Integration

, the integrand and bounds describe a solid capped by the elliptic paraboloid and lying above the xy-plane inside the unit circle. The request to 'change the order of integration' would apply if this were an improper triple integral being evaluated in Cartesian coordinates. Here, the order of integration (r, then θ) is itself the most simple and meaningful approach.

Learn more about Volume Calculation here:

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