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Properties of Definite Integrals A definite integral is a function that assigns a value to the area under a curve on a given interval are additive , linearity , continuity , monotonicity and upper and lowersums.

1. Additivity: The definite integral of a sum is equal to the sum of the definite integrals.

2. Linearity: The definite integral of a linear function is equal to the product of the function's slope and the length of the interval.

3. Continuity: The definite integral of a continuous function is equal to the area under the curve between two points.

4. Monotonicity: If a function is increasing or decreasing on a given interval, then the definite integral of the function is also increasing or decreasing.

5. Upper and Lower Sums: The upper and lower sums of a function are the sums of the areas of rectangles constructed under or over the curve between two points. The definite integral of the function is equal to the difference between the upper and lower sums.

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Answer:

Yes. Felix is correct.

Step-by-step explanation:

Find out how much ml does a can of lemonade have, I would assume its 330ml since I searched up the ml of a lemonade can.

Find out how much cans of lemonade there are in 2 packs of 8.

2 packs = 8 x 2 = 16 cans of lemonade

Now find out how much ml 16 lemonade cans equal to.

1 can of lemonade = 330ml

16 cans of lemonade = 330 x 16 = 5280ml

Now let's find 5 bottles of 1 litre.

1 litre x 5 = 5 litres.

5280 = 5 litres and 280ml

So Felix is correct about 2 packs of 8 cans is more than 5 bottles of 1 litre.

Answer:

The square root of 32 by 3 is 5.657

If that digit is less than 5, do not change the rounding digit but drop all digits to the right of it. If that digit is greater than or equal to five, add one to the rounding digit and drop all digits to the right of it.

Step-by-step explanation:

Answer:

8

Step-by-step explanation:

Answer:800% as a decimal = 8

Remove the percent sign, then divide 800 by 100:

800% → 800 → 800/100 = 8

Mr. Marshal spent 1/4 of his salary on miscellaneous expenses and $1,680 on rental; therefore, he spent 1/36 of his entire salary on the trip.

To find the fraction of money spent on miscellaneous, we need to calculate the sum of the fractions spent on rental, food, and bank and subtract it from 1.

Rental: 1/5

Food: 1/10

Bank: 1/4

To find the fraction spent on miscellaneous:

Fraction spent on miscellaneous = 1 - (Rental + Food + Bank)

Rental + Food + Bank = 1/5 + 1/10 + 1/4 = (8 + 4 + 5)/40 = 17/40

Fraction spent on miscellaneous = 1 - 17/40 = 23/40

So, Mr. Marshal spent 23/40 of his salary on miscellaneous.

To find the amount spent on rental, we multiply the fraction spent on rental by his salary:

Amount spent on rental = (1/5) \(\times\) $8,400 = $1,680

Therefore, Mr. Marshal spent $1,680 on rental.

If Mr. Marshal spent 4/9 of the miscellaneous amount on a trip, we need to calculate the fraction of his entire salary spent on the trip.

Fraction spent on the trip = (4/9) \(\times\) (23/40) = 92/360 = 23/90

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Help Center to Water Slide #1: Level line, y = 3. Toddler area to Concessions: Negative incline, y = (-1/3)x - 7. Gift Shop to Restrooms: Level line, y = -9. Security desk to Water Slide #2: Even line, y = 4. Lazy river to large Whirlpool: Positive incline, y = (2/5)x + 61/5.

To calculate the slope between attractions and discover the midpoints, we'll utilize the given requested sets for each fascination. Here are the calculations:

Help Center to Water Slide #1:

Slope = ((y2 - y1) / (x2 - x1)) = ((3 - 3) / (-6 - 2)) = / -8 =

Midpoint = ((x1 + x2) / 2), ((y1 + y2) / 2)) = ((2 + (-6)) / 2), ((3 + 3) / 2)) = (-2, 3)

Toddler area to Concessions:

Slope = ((y2 - y1) / (x2 - x1)) = ((-6 - (-3)) / (6 - (-3)) = (-3 / 9) = -1/3

Midpoint = ((x1 + x2) / 2), ((y1 + y2) / 2)) = ((-3 + 6) / 2), ((-9 + (-6)) / 2)) = (1.5, -7.5)

Gift Shop to Restrooms:

Slope = ((y2 - y1) / (x2 - x1)) = ((-9 - (-9)) / (8 - 2)) = / 6 =

Midpoint = ((x1 + x2) / 2), ((y1 + y2) / 2)) = ((8 + 2) / 2), ((-9 + (-9)) / 2) = (5, -9)

Security desk to Water Slide #2:

Slope = ((y2 - y1) / (x2 - x1)) = ((4 - 4) / (16 - 5)) = / 11 =

Midpoint = ((x1 + x2) / 2), ((y1 + y2) / 2)) = ((5 + 16) / 2), ((4 + 4) / 2) = (10.5, 4)

Lazy River to Large Whirlpool:

Slope = ((y2 - y1) / (x2 - x1)) = ((10 - 14) / (-7 - 3)) = (-4 / -10) = 2/5

Midpoint = ((x1 + x2) / 2)), ((y1 + y2) / 2)) = ((-7 + 3) / 2), ((10 + 14) / 2)) = (-2, 12)

Presently, let's calculate the linear equations for each case:

Help Center to Water Slide #1:

The Slope is 0, and the y-coordinate is consistent, so the equation is y = 3.

Toddler area to Concessions:

The Slope is -1/3, and the y-intercept can be calculated utilizing the midpoint facilitates:

-7.5 = (-1/3)(1.5) + b

-7.5 = -0.5 + b

b = -7

Gift Shop to Restrooms:

The Slope is 0, and the y-coordinate is steady, so the equation is y = -9.

Security desk to Water Slide #2:

The Slope is 0, and the y-coordinate is steady, so the equation is y = 4.

Lazy River to Large Whirlpool:

The Slope is 2/5, and the y-intercept can be calculated utilizing the midpoint arranges:

12 = (2/5)(-2) + b

12 = -4/5 + b

b = 61/5

The Linear equations for each case are as follows:

Help Center to Water Slide #1: y = 3

Toddler area to Concessions: y = (-1/3)x - 7

Gift Shop to Restrooms: y = -9

Security desk to Water Slide #2: y = 4

Lazy River to Large Whirlpool: y = (2/5)x + 61/5

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Answer:

Angle A = tan^-1(11/5)
Angle C = 90 - tan^-1(11/5)
AC = sqrt(146)

Step-by-step explanation:

As this is a right triangle, we can apply the Pythagorean theorem a^2 + b^2 = c^2, where c is the hypotenuse while a and b are the legs, to solve for AC.

11^2 + 5^2 = AC^2

121 + 25 = AC^2

146 = AC^2

sqrt(146) = AC^2

Next, to find angle A, we can use one of the trigonometric functions. Let’s use tangent for simplicity. Tangent of an angle is “opposite divided by adjacent”. If we set the angle to A, opposite is side BC and the adjacent is side AB. Thus, tan(A) = 11/5 and tan^-1(11/5) = A.
Since the sum of angles in a triangle is 180, we can find angle C by setting up this equation: C = 180 - 90 - tan^-1(11/5), which is 90 - tan^-1(11/5)

Answer: $144

Step-by-step explanation: To find the price she gets paid for 1 hr we need to divide the money by the hours. So, 90 / 5 = 18.

So, now we know she gets paid 18 dollars an hour. So, we need to find out how much she earns in 8 hrs. So, we would simply multiply the 18 by the 8, to get $144.

The sum of the interior angle of a rhombus is 360° and each angle is 90°

Given:

Points on the given function are (0,2), (1,0), and (2,2).

Required:

We need to find the piecewise function.

Explanation:

Consider the piecewise function.

The graph of the function is

The point lie on the function is (-2.2), (0,0) and (2,2).

The point (0,0) from the graph y=|x| is moved right one unit.

Reduce the values of x by 2 units since graph y=|x| is moved right one unit.

The difference between (-2,2) and (2,2) is (2-(-2))=4 units in graph y=|x|.

The difference between (0,2) and (2,2) is (2-0)=2 units in the given graph.

The given graph is shortened by 2 units.

Multiply the x coordinates by 2.

Final answer:

The y-coordinate of -3f(x-1) when x = 0 is -63

From the question, we have the following parameters that can be used in our computation:

The table of values

So, we have

-3f(x - 1)

When x = 0, we have

y = -3f(0 - 1)

This gives

y = -3f(-1)

From the table, we have

y = -3 * 21

Evaluate

y = -63

Hence, the y-coordinate is -63

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Answer:

18

Step-by-step explanation:

She won 75% of 24 matches.

75% of 24 = 0.75 × 24 = 18

Answer: 18

Tennis player will win 18 matches as a required percentage of total matches.

"Percentage is defined as the hundredth part of a given quantity."

According to the question,

Percentage of tennis player won the matches = 75%

Total number of matches played by player = 24

x represents the number of matches she win

Therefore,

   x = 75% of 24

⇒ x = (75 / 100) × 24

⇒ x = 18 matches

Hence, tennis player will win 18 matches.

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The capacity of this swimming pool is between B. 500 gallons to 1,000 gallons.

Capacity refers to the product of the length, width, and height of a three-dimensional object or space.

The capacity of an object means the same as its volume.

Olympic-size swimming pools have the following standard dimensions:

Length = 50 m

Width = 25 m

Height = 2 m

Capacity of an Olympic swimming pool = 2,500 m³ (50 x 25 x 2)

= 660,000 gallons (2,500 x 1,000 ÷ 3.785)

An interval estimate shows the lower and upper limits.

6 x 105 gallons = 630 gallons

Thus, the capacity of the swimming pool is Option B.

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A. 100 gallons to 500 gallons

B. 500 gallons to 1,000 gallons

C. 1,000 gallons to 1,500 gallons

D. 1,500 gallons to 2,000 gallons

The most suspicious set of outcomes of 30 games played between the two computers would be ABBABABABABBABABABABABABABABAA.

In a fair game where player B is supposed to win 4 out of 5 times, we would expect player B to win approximately 24 out of 30 games.  However, option C shows player B winning only 6 out of 30 games, which deviates significantly from the expected outcome.

This set of outcomes suggests a strong bias towards player A winning, as player B's wins are significantly lower than expected.

It is highly unlikely to observe such a low success rate for player B if the game was truly biased in favor of player B winning 4 out of 5 times.

Option C raises suspicions of an unfair advantage given to player A, potentially indicating some manipulation or incorrect implementation in the programming of the game.

Further investigation would be warranted to determine the cause of this biased outcome and to ensure fair play between the two computers in future games.

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If the random variable x is normally distributed with a mean equal to 0.45 and a standard deviation equal to 0.40, then P(x ≥ .75) is 0.9227.

A z-score describes the position of a raw score in terms of its distance from the mean when measured in standard deviation units. The z-score is positive if the value lies above the mean and negative if it lies below the mean.

Given the problem above, we need to find what the z-score is when P(x ≥ .75).

The formula for calculating a z-score is given by:

\(Z=\dfrac{\text{x}-\mu}{\sigma}\)

Where:

Now,

\(Z=\dfrac{\text{x}-\mu}{\sigma}\)

\(Z=P(\text{x} \geq 0.75) = \huge \text(\dfrac{P(Z \geq (0.75 - 0.45)}{0.40}\huge \text)\)

\(Z= P\huge \text (\dfrac{Z \geq0.30}{0.40}\huge \text)\)

\(Z=P(Z \geq 0.75) = 1 - P(Z < 0.75)\)

\(Z=1 - 0.077337\)

\(Z\thickapprox 0.9227\)

Therefore, the z-score of P(x ≥ .75) is 0.9227.

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The roof top would be 0. At 12 seconds the ball is at -225.6, which would be the ground so the building would be 225.6 meters tall.

The highest positive value is 81.6, so this is the highest the ball goes up.

The value changes from a positive to a negative between 8 and 10, which means that would be around the time it is the same height as the building.

The answers would be:

The first and third choices

By the Pythagorean theorem,

\(\sin^2 t +\cos^2 t = 1 \\ \\ \sin^2 t +(-3/4)^2 = 1 \\ \\ \sin^2 t =7/16 \\ \\ \sin t=-\frac{\sqrt{7}}{4}\)

(I took the negative case because of the range of values of t)

Using the cosine double angle formula,

\(\cos 2t=2\cos^2 t -1 =2(-3/4)^2 - 1 =1/8\)

Using the sine double angle formula,

\(\sin 2t=2\left(-\frac{3}{4} \right)\left(-\frac{\sqrt{7}}{4} \right)=\frac{3\sqrt{7}}{8}\)

Using the cosine half-angle formula,

\(\cos \frac{t}{2}= -\frac{\sqrt{1+\cos A}}{2}=-\frac{\sqrt{1+\frac{3\sqrt{7}}{8}}}{2}\)

(I took the negative case because of the range of values of t)

Using the sine half-angle formula,

\(\sin \frac{t}{2}=\frac{\sqrt{1-\cos A}}{2}=\frac{\sqrt{1-\frac{3\sqrt{7}}{8}}}{2}\)

(I took the positive case because of the range of values of t)

Answer:

9 ft

Step-by-step explanation:

Area of a a rectangle = length × width.

Plug in the information youhave.

45ft. = 5ft × ?

45 = 5x

9 = x

Answer:

the answer is 11

I hope it helps

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Answer:

30m

Step-by-step explanation:

To find area of triangle you do half the base times the height. The base is 5 so half of that is 2.5. The height is 12. So, 12 times 2.5 equals 30.

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Answer:

I believe the equation is Area= ½ base x height

Step-by-step explanation:

Area= ½ 13 x 5

Area= 6.5 x 5

Area= 32.5 m

Answer:

\(x=5\)

Step-by-step explanation:

The perimeter of a figure is the sum of all its sides.

The sides are x, (x+3), and (2x). The perimeter is 23, so the sum of the expression equates to 23. Thus:

\(x+(x+3)+(2x)=23\)

Combine like terms:

\(4x+3=23\)

Subtract 3 from both sides:

\(4x=20\)

Divide both sides by 4:"

\(x=5\)

And we're done!

Answer:

(-x,y) To (x,-y)

Step-by-step explanation:

the first distribution is symmetric and has a bell shape. And the second one is skewed to the right

The number of times Jennifer rolls a number less than 3 in a number cube when she rolls it 60 times is 40.

The probability of an event is the ratio of number of favorable outcome to the total number of outcome of that event.

The probability of rolling a number less than 3 on a number cube is 2/6.

\(P( > 3)=\dfrac{2}{6}\)

Here, the Jennifer rolls a number cube 60 times. Thus, the number of times she rolled the number less than 3 in this cube is,

\(n=\dfrac{2}{3}\times60\\n=40\)

Thus, the number of times Jennifer rolls a number less than 3 in a number cube when she rolls it 60 times is 40.

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Thus, the scale factor from circle A to circle B is found to be 2.

The ratio between comparable measurements of just an element and a portrayal of that element is known as a scale factor in mathematics. The copy will be larger if the scaling factor is a whole number. A fractional scaling factor means that the duplicate will be smaller.

You must first choose which direction we are scaling in order to determine the scale factor:

The radius of circle A = 2 units

The radius of circle B = 4 units.

So, there is a dilation with the scale factor of 2 units as 2*2 = 4 units.

Thus, the scale factor from circle A to circle B is found to be 2.

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Answer:

Step-by-step explanation:

To find the linearization of the function f(x) = sqrt(x) at x = a, we can use the linearization formula:

f(x) ≈ f(a) + f'(a) (x - a)

To use this formula, we first need to find the value of f(a) and f'(a). At x = a = 4, the value of f(x) is f(4) = sqrt(4) = 2. The derivative of f(x) is f'(x) = 1/(2 * sqrt(x)), so the value of f'(a) is f'(4) = 1/(2 * sqrt(4)) = 1/4.

Substituting these values into the linearization formula, we get:

f(x) ≈ 2 + 1/4 (x - 4)

This is the linearization of f(x) = sqrt(x) at x = 4. It is an approximation of the function f(x) that is valid for values of x that are close to 4.

Answer:

We can use the formula for compound interest to find the future value (FV) of Natalie's investment:

FV = P * (1 + r/n)^(n*t)

Where:

P is the principal amount (the initial investment), which is $2,000 in this case

r is the annual interest rate as a decimal, which is 11% or 0.11

n is the number of times the interest is compounded per year, which is 12 since interest is compounded monthly

t is the number of years, which is 20

Substituting the values into the formula, we get:

FV =

2

,

000

∗

(

1

+

0.11

/

12

)

(

12

∗

20

)

�

�

=

2,000 * (1.00917)^240

FV = $18,255.74

Therefore, after 20 years of compounded monthly interest at a rate of 11%, Natalie's investment of 2,000 will be worth approximately 18,256.

Answer:

$17,870

Step-by-step explanation:

You must use the formula for compound interest.

A = P(1 + r/n)^nt

I suggest you look it up at some point so that you can do these more easily in the future!

a) The probability that exactly one of them will be a girl = 0.3407

b) The probability that at least one of them will like the football = 0.7672

a) If we select three students then the probability that exactly one of them will be a girl

From the attached two way table we can observe that the total number of girls = 22

the total number of boys = 18

and the total number of students = 40

The possible outcomes for selecting 3 students from 40 would be,

⁴⁰C₃

Using combination formula,

⁴⁰C₃ = 40! / (3! × (40 - 3)!)

      = 9880

If there is exactly one girl then other two must be boys in the set of 3 selected students.

So, the required probability would be,

P = (²²C₁ × ¹⁸C₂) / ⁴⁰C₃

P = (22 × 153)/9880

P = 0.3407

b) The number of students like the football = 15

and the number of students who don't like the football are 40 - 15 = 25

The probability that at least one of them will like the football would be,

P = (¹⁵C₃ × ²⁵C₀ + ¹⁵C₂ × ²⁵C₁ + ¹⁵C₁ × ²⁵C₂) / ⁴⁰C₃

P = ((455 × 1) + (105 × 25) + (15 × 300)) / 9880

P = 0.7672

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Answer:

B) y - 2 = (-1/4)(x - 2)

Step-by-step explanation:

Start with the equation of the given line: y = 4x + 7.

Determine the slope of the given line, which is 4.

Since the perpendicular line has a negative reciprocal slope, the perpendicular line's slope is -1/4.

Use the point-slope form of a linear equation: y - y₁ = m(x - x₁), where (x₁, y₁) is the given point and m is the slope.

Substitute the values x₁ = 2 and y₁ = 2 into the point-slope equation: y - 2 = (-1/4)(x - 2).

Simplify by distributing -1/4 to (x - 2): y - 2 = (-1/4)x + 1/2.

Add 2 to both sides to isolate y: y = (-1/4)x + 1/2 + 2.

Simplify further: y = (-1/4)x + 5/2.

The equation in point-slope form for the line passing through (2, 2) and perpendicular to y = 4x + 7 is y - 2 = (-1/4)(x - 2), which matches option B.