2. consider the trapezoid in the previous problem. locate the center of mass of the trapezoid. remember to again break the trapezoid into horizontal or vertical strips.

The value of the definite integral is 10 after breaking the limits of the integral.

It is defined as the mathematical approach to calculating the smaller parts or components.

It is given that:

The piecewise function is given:

[tex]\rm \int\limits^4_0 {x} \, dx = \int\limits^2_0 {2} \, dx +\int\limits^4_2 {x} \, dx[/tex]

[tex]\rm \int\limits^4_0 {x} \, dx = x^2|^2_0+(1/2)x^2|^4_2[/tex]

= 4 + (1/2)[16 - 4)

= 4 + 6

= 10

Thus, the value of the definite integral is 10 after breaking the limits of the integral.

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Answer:

Step-by-step explanation: y = -3x + 6

The coordinates of the midpoint are (-1, 3.5) and the distance of AC is 26.25 units.

We know that for a segment whose endpoints are (x₁, y₁) and (x₂, y₂), the coordinates of the midpoint are:

( [x₁ + x₂]/2, [y₁ + y₂]/2)

In this case, the endpoints are:  (3, -9) and (-5, 16), then the midpoint is:

M = ( [3 - 5]/2, [-9 + 16]/2)

M = (-1, 3.5)

The distance between two points (x₁, y₁) and (x₂, y₂) is:

d = √( (x₁ - x₂)^2 + (y₁ - y₂)^2)

So here the distance is:

d = √( (3 + 5)^2 + (-9 - 16)^2) = 26.25 units.

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The required Interest Rate Compounded Daily​ (365 Days/Year) is 7.227%.

Interest on interest, or compound interest, is the adding of interest to the principle sum of a loan or deposit. It's the outcome of reinvesting interest rather than paying it out, so that interest is received on the principal plus previously collected interest in the next quarter.,

[tex]A = P(1+ \dfrac{r}{n})^{nt}[/tex]

where A is the final amount

P is the principal amount

r is the rate of interest

n is the number of times interest is charged in a year

t is the number of years

Given that interest Rate Compounded Daily​ (365 Days/Year) is required for a ​$23,000 Investment to grow To $35,500 In 6 ​Years. Therefore, we can write,

[tex]A = P(1+\dfrac rn)^{n \times t}[/tex]

[tex]35,500 = 23,000(1+\dfrac{r}{365})^{365 \times 6}\\\\\dfrac{35,500}{23,000} = (1+\dfrac{r}{365})^{2190}\\\\1.543478 = (1+\dfrac{r}{365})^{2190}\\\\1.000198 = 1+\dfrac{r}{365}\\\\\dfrac{r}{365} = 1.000198 - 1\\\\\dfrac{r}{365} = 0.000198[/tex]

r = 1.000198 x 365

r  = 0.07227

r = 7.227%

Hence, the required Interest Rate Compounded Daily​ (365 Days/Year) is 7.227%.

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The average score is 97

What is average ?

In plain English, an average is a single number chosen to represent a group of numbers; it is often the sum of the numbers divided by the number of numbers in the group. The average of the numbers 2, 3, 4, 7, and 9 is, for instance, 5.

Average, mean, median, and norm all refer to something that sits in the middle. The average is the ratio created by dividing the total number of figures in a set by the total number of figures. scored 85 on tests on average. The term "mean" can refer to the simple average or to a value that lies halfway between two extremes.

Let the grade he got on final test be x

The average of all four grades is 91.

So, we have  

(92+89+86+x)/4 = 91

⇒x=364−267

= 97

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Answer: Nancy ran, 14080 yards.

Step-by-step explanation:

15 different combinations can be used by John to buy the cake and the hammer.

Given,

John buys a cake at a bakery and a hammer at a hardware store

Number of hardware shops = 5

Number of bakeries = 3

We have to find the combinations of stores he can purchase;

Here,

5 hardware shops; H₁, H₂, H₃, H₄, H₅

3 bakeries; B₁, B₂, B₃

Now,

The combinations be like;

B₁ ; (H₁, H₂, H₃, H₄, H₅)

B₂ ; (H₁, H₂, H₃, H₄, H₅)

B₃ ; (H₁, H₂, H₃, H₄, H₅)

That is,

3 bakeries and 5 hardware shops

= 3 x 5

= 15

Therefore,

John can buy the cake and hammer in 15 different combinations.

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When the standard deviation is $1200 and the mean is $8000, the test statistic's result will be -1.804.

A statistic employed in statistical hypothesis testing is known as a test statistic. A test statistic, which can be thought of as a numerical summary of a data set that condenses the data into a single number that can be used to conduct the hypothesis test, is how a hypothesis test is commonly expressed. A number derived from a statistical test of a hypothesis is the test statistic. It displays how closely your actual data fit the distribution predicted by the statistical test's null hypothesis.

Here,

The claimed mean=$8250

The actual mean=$8000

The sample size=75

The standard deviation=$1200

The test statistic,

=(Actual mean-claimed mean)/Standard deviation/√(sample size)

=(8000-8250)/1200/√75

=-250/1200*1/√75

=-1.804

The value of test statistic will be -1.804 when the mean is $8000 and standard deviation is $1200.

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Answer:

There are 2 stars

Step-by-step explanation:

There are 29 items in total.

As we are given the number of two of the items(nutcrackers and snowmen)we can easily find the number of the last item(stars) by subtracting the number of nutcrackers and snowman by the number of total items.(29)

So the solution would be :

29 - (13 + 14) = number of stars

==> add 13 and 14

29 - 27 = number of stars

==> simplify

2 = number of stars

The pmf of X = the number of children in the family is

1/4 + 3/8 + 3/8 =1

It is given that,

a family decides to have children until it has three children of the same gender i.e. BBB or GGG.

X can take the values {3,4,5}

and we know that

P(X = 3) + P(X = 4) + P(X = 5) = 1

We will find each term one by one,

P(X = 3) = ([tex]\frac{1}{2} * \frac{1}{2} * \frac{1}{2}[/tex])*2

             = [tex]\frac{1}{8}[/tex] * 2

P(X = 3) = [tex]\frac{1}{4}[/tex]

P(X = 4) = ([tex]\frac{1}{2} * \frac{1}{2} * \frac{1}{2} * \frac{1}{2}[/tex])*6

             = ([tex]\frac{1}{16}[/tex]) * 6

P(X =4)  = [tex]\frac{3}{8}[/tex]

P(X = 5) = ([tex]\frac{1}{2} * \frac{1}{2} * \frac{1}{2} * \frac{1}{2} * \frac{1}{2}[/tex])*12

             = ([tex]\frac{1}{32}[/tex])* 12

P(X = 5) = [tex]\frac{3}{8}[/tex]

Therefore the pmf equation implies,

[tex]\frac{1}{4} + \frac{3}{8} + \frac{3}{8}[/tex] = [tex]\frac{32}{32}[/tex] = 1.

The probability mass function (pmf) is also called a probbility function or frequency which characterizes the distribution of a discrete random variable. It is a function over the sample space of a discrete random variable X which gives the probability that X is equal to a certain value.

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Answer:

Step-by-step explanation:

Answer:

(10) x=1, y=-3

(12) x=3, y=22

(13) x= 6, y= 7

Step-by-step explanation:

[10] 6x -9 =-3x

6x+3x=9

9x=9

x=9,

The volume of the vanilla ice cream in the cone is 4.18 cubic inches

The diameter of the cone = 2 inches

The radius of the cone = 2/1

= 1 inches

The height of the cone = 6 inches

The volume of the cone = (1/3) × π × r^2 × h

Substitute the values in the equation

= (1/3)π × 1^2 × 6

= 6.28 cubic inches

The ratio of  vanilla and chocolate ice cream = 2:1

2x + x = 6.28

3x = 6.28

x = 6.28 / 3

x = 2.09

The volume of vanilla ice cream = 2x

= 2×2.09

= 4.18 cubic inches

Therefore, the volume of vanilla ice cream is 4.18 cubic inches

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The only option that represents a proportional relationship is;

d) The cost of repairing a computer at $54.75 per hour

Proportional relationships are defined as relationships between two variables where their ratios are equivalent.

a) The cost of purchasing boxes of paper is $25.39 per box plus a shipping fee of $3.25. If the number of boxes are x, then;

y = 25.39x + 3.25

This does not represent a proportional relationship as their ratios are not equivalent.

b) The cost of repairing a copy machine is is $35.75 per hour with a service charge of $80.50. The equation is;

y = 35.75x + 80.50

This does not represent a proportional relationship as their ratios are not equivalent.

c) The cost of purchasing printer cartridges at $34.85 per cartridge including delivery. This is not a proportional relationship as the ratio is not a straightforward one based on the delivery.

d) This represents a proportional relationship because the equation is;

y = 54.75x

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The complete options are;

a) The cost of purchasing boxes of paper for $25.39 per box plus a shipping fee of $3.25 per box .

b) The cost of repairing a copy machine at $35.75 per hour with a service charge of $80.50

c) The cost of purchasing printer cartridges at $34.85 per cartridge including delivery

d) The cost of repairing a computer at $54.75 per hour

Answer:

seven and seven hundred nintey three thousandths

Step-by-step explanation:

Step-by-step explanation:

It is not a scalene triangle

because the third angle is 69⁰

Explanation: 180-(69+42)

180-111

69

There are 55 sophomores taking French II and Chemistry I.

In mathematics, Multiplication operations perform Multiplying values on either side of the operator.

For example 4×2 = 8

According to the given information, the required solution would be as:

First, we need to find the number of sophomores taking French II, Chemistry  I, or both:

⇒ 1/3 × 465

Apply the multiplication operation, and we get

⇒ 155

Of the 155 sophomores, 40 are taking only French II and 60 are taking only Chemistry I:

⇒ 155 - 40 - 60

⇒ 155 - 100

Apply the subtraction operation, and we get

⇒ 55

Thus, there are 55 sophomores taking French II and Chemistry I.

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Answer: The second one (B)

Hope this helps! :)

Step-by-step explanation:

The times when Hayden made the bracelets in this situation don't matter. All that matters in this problem is how many he made. Since it tells you that he made them for 5 days, you would multiply the number of bracelets by the days, to get the total. (Or write the equation like so: 5*(2+4))

And of course, since they are divided into groups as the last step, we have division.

5*(2+4) /6

Answer:

[tex]-\frac{4}{51}\approx-0.07843[/tex]

Explanation:

[tex]\frac{4}{15}(-\frac{1}{3.4})\\\frac{4}{15}(-\frac{5}{17})\\-\frac{20}{255}\\-\frac{4}{51}\approx-0.07843[/tex]

Answer:

slope = 1/8

equation: y = 1/8 x

Step-by-step explanation:

Let distance = y.

Let time = x.

y = mx + b

Look at point (1, 8).

Also, the line passes through (0, 0).

slope = m = 1/8

Since it is a direct proportion, b = 0.

Equation: y = 1/8 x

The ancient stone pyramids in Egypt, which have a square base with a width of 148 meters and a height of 126 meters, have a volume of 9394 square metres.

Mathematics uses cubic units to quantify volume, including cubic meters, cubic centimeters, cubic millimeters, etc. Typically, liters and gallons are used to calculate liquid volume. The loudness of a sound is measured by its volume. A container's capacity, or how much it can hold, would be determined by its volume. Cubic units established by the International System of Units are frequently used to express volume. Learn how to easily measure volume.

Here,

Side of the pyramid=148 m

Height of pyramid=126 m

Volume,

=1/2*b*h

=1/2*148*126

=9394 m²

The volume of the ancient stone pyramids in Egypt that has a square base of side 148 m and height 126m is 9394 m².

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The probability that you have seen both the red and green marbles at least once is 0.3104

Given,

1/5 chance of winning a marble of any color in a drawing.

Every draw is independent and has the same probability because the marbles are replaced.

Calculating = 1- the likelihood of not receiving both red and green marbles in any of the four draws yields the same result as P, the probability of receiving both red and green marbles at least once (Q).

The union of the following three examples represents ways to not receive both red and green in all four draws:

(1)

Probability of no red and no green in a draw = 3/5.

Probability of no red and no green in 4 draws,

Q1 = 3/5 * 3/5 * 3/5* 3/5 = 81/54

The probabilities are multiplied as each draw is independent and has the same probabilities.

(2)

The probability of no red and 1 green in 4 draws is given by binomial probability.

= (4 3) (1/5)(3/5)³ = (4 * 3³)/5¹ = 108/5⁴

The probability of no red and 2 green in 4 draws is given by binomial probability.

= (4 2) (1/5)²(3/5)² == (6 * 3²)/54 = 54/5⁴

The probability of no red and 3 green in 4 draws is given by binomial probability.

= (4 3) (1/5)³(3/5) == (4 * 3)/5¹ = 12/5⁴

The probability of no red and 4 green in 4 draws is given by binomial probability.

= (4 4) = 1 * 1/5¹ = 1/5⁴⁴

Q2= sum of all 4 above probabilities.

Q2 (108 +54 + 12+1)/5 = 175/5⁴

(3)

This probability is the same as in Q2. Q3 = 175/5⁴

Therefore the probability of not seeing both red and green at all,

Q = Q1 + Q2 + Q3.

   = 81 +175 + 175/5⁴ = 413/5⁴

Finally, the probability of getting both red and green at least once,

P = 1 - Q

  = 1-413/5⁴

  = (625 431)/625

  = 194/625

  = 0.3104.

The answer is 0.3104.

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Using the z-score a randomly selected Endure All battery has a 10.56% chance of lasting more than 550 hours.

How closely a value relates to the mean of a set of values is measured using a Z-score.

Z-score is calculated using standard deviations from the mean.

When a data point's Z-score is 0, it indicates that its entire score is equal to the mean score.

So, how far the raw score deviates from the mean is determined using the Z score.

These steps are used to determine the Z score:

z = (x - μ)/σ

We know that:  x > 550 and μ = 500, σ = 40

z = 550 - 500 / 40

z = 1.25

Using the normal distribution graph as a guide:

P(z > 1.25) = 1 - P(z < 1.25)

P(z > 1.25) = 1 - 0.8944

P(z > 1.25) = 0.1056

Therefore, using the z-score a randomly selected Endure All battery has a 10.56% chance of lasting more than 550 hours.

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Using the probability formula, the number of diamonds needed to make the game fair is determined by option (B) 6.

Assume that there are a finite number of elementary events that are all equally likely to occur in the sample space of the experiment under consideration.

So let's say we want to determine the likelihood of an occurrence E.

So, the value of diamonds has to be:

The expected number of points that one can lose should equal the expected number of points that one can acquire for the game to be fair.

So, in order for the game to be fair, we need:

Expected points earned by selecting a diamond minus anticipated points lost by selecting a diamond.

If we choose a diamond, let's say we obtain P points.

Let X represent the random variable that records the player's score for each individual draw.

13/52 = 1/4 is the probability of selecting a diamond from a deck of 52 cards.

This is due to the fact that there are only 13 ways to choose a diamond out of the total 52 ways to choose a card.

So, the likelihood of receiving P points is 1/4, or 0.25.

The odds of not getting a diamond equal 1 minus the odds of getting a diamond equal 1 minus 0.25, which equals 0.75.

The points we receive when we don't acquire a diamond are -2. (negative since we actually lose it, so we can take it as a negative gain).

E(X) = expectation of X = 0 is therefore required (since the negative points of loosing would equate to positive point of winning, thus, making the expected overall score 0, thus, keeping it a fair game.)

We now have:

When choosing a non-diamond card, P(X = -2) equals 0.75; when choosing a diamond card, P(X = p) equals 0.25.

There are only possible values for X.

The result is:

E(X) = -2*p(X=-2)+p*p(X=p)

E(X) = -2*0.75+p*0.25

0=-1.5+0.25p

p = 1.5/0.25 = 6

Therefore, using the probability formula, the number of diamonds needed to make the game fair is determined by option (B) 6.

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Complete question:
You pick a card at random from an ordinary deck of 52 cards. If the card is a diamond, you get some points; if not, you lose 2 points. What value for the diamonds would make the game fair?

a) 12

b) 6

c) 24

d) 3

Answer:

B-6

Step-by-step explanation: