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The state of Colorado had roughly the shape of a rectangle that is 3.8 x 10^2

2.8 x 10^2 miles high. What is the approximate area of Colorado? hint: the area of the rectangle is a product of its length and width.

Final Answer:

The maximum value of f(x) = xa(7 - x)b on the interval 0 ≤ x ≤ 7 occurs when x = 7/2 for a > 0 and b > 0.

Explanation:

To find the maximum value of f(x) on the interval 0 ≤ x ≤ 7, we'll use calculus. First, take the derivative of f(x) with respect to x.

f'(x) = a * (7 - x)^b * x^(a-1) - b * a * (7 - x)^(b-1) * x^a

Setting f'(x) = 0 to find critical points:

a * (7 - x)^b * x^(a-1) - b * a * (7 - x)^(b-1) * x^a = 0

Solving for x gives x = 7/2. To determine if this point is a maximum, minimum, or inflection point, we'll use the second derivative test.

f''(x) = a * (7 - x)^(b-1) * x^(a-2) * ((a - 1) * x - 7) - b * a * (7 - x)^(b-2) * x^(a-1) * ((b - 1) * x - 7)

When x = 7/2, f''(7/2) > 0, indicating a concave-up shape and confirming it as a local minimum. Therefore, the maximum value of f(x) occurs at the endpoints of the interval, x = 0 and x = 7, or potentially at x = 7/2 if it surpasses these values.

For a > 0 and b > 0, plugging in x = 7/2 into f(x) = xa(7 - x)b yields the maximum value of the function within the given interval.

Understanding the critical points through derivatives helps identify where the maximum or minimum points occur in the function. In this case, x = 7/2 stands as the point of maximum value within the interval 0 ≤ x ≤ 7 when a and b are both positive.

The area of the triangle is 6 square units.

To find the area of a triangle with vertices at (-2, 1), (2, 1), and (3, 4), we can use the formula for the area of a triangle: Area = 1/2 × base × height.

First, we need to find the base and height of the triangle. The base is the distance between the points (-2, 1) and (2, 1), which is 4 units. The height is the distance between the point (3, 4) and the line formed by the base, which is the vertical distance from (3, 4) to the line y=1, which is 3 units.

Plugging these values into the formula, we get Area = 1/2 × 4 ×3 = 6 square units.

Answer:

answer is 1440,ok

Step-by-step explanation:

from x15 to x 25 is 4 - -16 = -20 degree change

25 -15 = 10000 feet

-20/10 = -2 degrees every 1000 feet

Answer:

I hope this helps

Step-by-step explanation:

The probability of selecting 22 red balls is  [tex]\frac{^{44} C_{22} }{^{99} C_{22}}[/tex]

It is a branch of mathematics that deals with the occurrence of a random event.

Given,

A bag contains 44 red balls and 55 blue balls

The total number of balls=44+55=99

22 balls are selected at random from bag.

We need to find the probability of selecting 22 red balls.

The probability of selecting 22 red balls from 44 balls

[tex]^{44} C_{22}[/tex]

2,104,098,963,720

and [tex]^{99} C_{22}[/tex] is 5.7190121704386E+21

So probability of selecting 22 red balls is

[tex]\frac{^{44} C_{22} }{^{99} C_{22}}[/tex]

Hence, the probability of selecting 22 red balls is  [tex]\frac{^{44} C_{22} }{^{99} C_{22}}[/tex]

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the answer would be 37

Answer:

a) [tex]4\left(3x+2\right)-2=12x+6[/tex]

b) [tex]20b-16= 4\left(5b-4\right)[/tex]

Step-by-step explanation:

a) To simplify the expression you must:

Expand [tex]4(3x + 2)[/tex]:

Apply the distributive law: [tex]\:a\left(b+c\right)=ab+ac[/tex]

[tex]a=4,\:b=3x,\:c=2\\\\4\left(3x+2\right)=4\cdot \:3x+4\cdot \:2[/tex]

Simplify [tex]4\cdot \:3x+4\cdot \:2= 12x+8[/tex]

[tex]12x+8-2[/tex]

Subtract the numbers: [tex]4\left(3x+2\right)-2=12x+6[/tex]

b) To factor the expression [tex]20b - 16[/tex] you must:

Rewrite 16 as [tex]4\cdot 4[/tex] and 20 as [tex]4\cdot 5[/tex]

[tex]20b-16=4\cdot \:5b-4\cdot \:4[/tex]

Factor out common term 4

[tex]20b-16= 4\left(5b-4\right)[/tex]

Answer:  The required sequence of transformations is

Transformation 1 : a rotation by 90 degrees in the counterclockwise direction about the origin, (x, y) ⇒  (-y, x).

Transformation 2 : a translation by 2 units right and 3 units down, (x, y) ⇒ (x+2, y-3).

Step-by-step explanation:  We are given to describe in words a sequence of transformations that maps ∆ABC to ∆A'B'C'.

From the figure, we note that

the co-ordinates of the vertices of triangle ABC are A(0, 4), B(0, 0) and C(2, 3).

And, the co-ordinates of the vertices of triangle A'B'C' are A'(-2, -3), B'(2, -3) and C'(-1, -1).

We see that if triangle ABC is rotated 90 degrees in anticlockwise direction about the origin, then its co-ordinates changes according to the following rule :

(x, y)  ⇒  (-y, x).

That is

A(0, 4)  ⇒  (-4, 0),

B(0, 0)   ⇒ (0, 0),

C(2, 3)    ⇒ (-3, 2).

Now, if the vertices of the rotated triangle are translated  2 units right and 3 units down, then

(x, y)  ⇒  (x+2, y-3).

That is, the final co-ordinates after rotation and translation will be

(-4, 0)  ⇒  (-4+2, 0-3) = (-2, -3),

(0, 0)   ⇒ (0+2, 0-3) = (2, -3),

(-3, 2)   ⇒ (-3+2, 2-3) = (-1, -1).

We see that the final co-ordinates are the co-ordinates of the vertices of triangle A'B'C'.

Thus, the required sequence of transformations is

Transformation 1 : a rotation by 90 degrees in the counterclockwise direction about the origin, (x, y) ⇒  (-y, x).

Transformation 2 : a translation by 2 units right and 3 units down, (x, y) ⇒ (x+2, y-3).

To solve for "p" in the equation "t = d + pm," isolate "p" by subtracting "d" from both sides: "t - d = pm." Then, divide both sides by "m" to find "p": "p = (t - d) / m." This formula allows you to calculate "p" based on given values of "t," "d," and "m."

To solve for "p" in the equation "t = d + pm," you need to isolate "p" on one side of the equation. Here's how you do it step by step:

Start with the Equation:

The equation is t = d + pm, where "t" represents a value, "d" represents another value, "p" is the variable we want to solve for, and "m" is yet another value.

Isolate the "pm" Term:

To get "pm" by itself, we need to eliminate the "d" term on the right side. We can do this by subtracting "d" from both sides of the equation:

t - d = pm

Solve for "p":

Now that "pm" is isolated on the right side, we can solve for "p" by dividing both sides by "m":

p = (t - d) / m

So, the solution for "p" in terms of "t," "d," and "m" is given by the formula p = (t - d) / m. This formula tells you how to calculate the value of "p" based on the values of "t," "d," and "m."

In practical terms, this equation might be used in various scenarios. For example, if "t" represents total distance, "d" represents a fixed distance, and "m" represents a constant speed, then "p" would represent the time it takes to cover the remaining distance to reach "t." This formula can be quite useful in solving real-world problems involving time, distance, and speed.

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Answer:  The required probability of getting two green marbles is 49.45%.

Step-by-step explanation:  Given that a bag contains 10 green marbles and 4 yellow marbles. Two marbles are chosen at random, one at a time and without replacement.

We are to find the probability of getting two green marbles.

Let S denote the sample space for the experiment of choosing a marble from the bag and A denote the event of getting a green marble.

The, n(S) = 10 + 4 = 14   and   n(A) = 10.

So, the probability of event A will be

[tex]P(A)=\dfrac{n(A)}{n(S)}=\dfrac{10}{14}=\dfrac{5}{7}.[/tex]

After getting one green marble and not replacing, let S' denote the sample space for the experiment of choosing a marble from the bag

and

let B denote the event of getting another green marble.

Then, n(S') = 14 - 1 = 13    and    n(B) = 10 - 1 = 9.

Then, the probability of getting two green marbles is given by

[tex]P\\\\=P(A)\times P(B)\\\\=\dfrac{5}{7}\times\dfrac{n(B)}{n(S')}\\\\\\=\dfrac{5}{7}\times\dfrac{9}{13}\\\\\\=\dfrac{45}{91}\times100\%\\\\=49.45\%.[/tex]

Thus, the required probability of getting two marbles is 49.45%.

Answer:

4.5 hours

[tex]496 x= 558 x-279[/tex]

Step-by-step explanation:

We are given that two airplanes left the same airport and arrives at the same destination at the same time.

We have to find the number of hours taken by first plane to travel to the destination and find the equation that can be used to solve this problem

We are given that the first airplane left at 8:00 a.m

Let y be the distance traveled by the first airplane

The average rate of first airplane =496 miles per hour

The average rate of second airplane =558 miles per hour

Let x represents the number of hours that the first airplane traveled.

The number of hours that the second plane traveled =x-0.5

Because the second airplane take half an hour less than the first airplane

We know that [tex] distance=speed\times time [/tex]

[tex]d=496 x[/tex]

[tex]d=558(x-0.5)[/tex]

[tex]496 x= 558 x-279[/tex]

[tex]279=558 x-496 x[/tex]

[tex]279=62 x[/tex]

[tex]x=\frac{279}{62}[/tex]

x=4.5 hours

Hence, the first airplane takes 4.5 hours to travel to the destination.

Let's suppose the breadth of the rectangle be x cm 

The length of the rectangle = x + 3 cm

It is assumed that the length and breadth of the rectangle is amplified by 2.

Now, the breadth of the rectangle = x + 2

The length of the rectangle = x + 3 + 2 = x + 5

It is also assumed that the area of the rectangle is 70 sq. cm

Hence, the area of the rectangle = length x breadth = (x + 2)(x + 5) = 70

= x2 + 5x + 2x + 10 = 70

= x2 + 7x = 70 - 10

=  x2 + 7x = 60

= x2 + 7x - 60 = 0

= x2 + 12x - 5x - 60 = 0

=  x( x + 12) -5(x + 12) = 0

= ( x + 12) (x - 5) = 0

=x + 12 = 10               or              =x - 5 = 0

= x = 10 - 12 = -2               = x = 5

The breadth of a rectangle is never negative, so the value of x is 5 cm

Therefore, the breadth of the rectangle = 5 cm

The length of the rectangle = 5 + 3 = 8 cm