Parametric Functions

The probability that the average length of a randomly selected bundle of steel rods is greater than 168.58 cm is 0.7486

First, we need to calculate the z-score for the sample mean:

z = (x - μ) / (σ / √n)

z = (168.58 - 169) / (2.2 / √10)

z = -0.67

Next, we need to find the probability that a randomly selected bundle of steel rods will have a mean length greater than 168.58 cm. This is equivalent to finding the area under the standard normal distribution curve to the right of z = -0.67.

Using a standard normal distribution table or calculator, we find that the probability is 0.7486.

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A. The 51st note on a piano keyboard corresponds to a pitch of 440 cycles per second.

B. The pitch that is 73 notes higher on the keyboard has a frequency of about 1760 cycles per second.

A. Use the formula to find how many notes up the piano keyboard the pitch of 440 cycles per second corresponds to:

440 = 27.5 × 2⁽ⁿ⁻¹⁾/12

Dividing both sides by 27.5 and taking the logarithm with base 2 gives:

log₂(440/27.5) = (n-1)/12

n-1 = 12 × log₂(440/27.5)

n-1 = 12 × 4.1702 ≈ 50.042

n ≈ 51.042

Therefore, the pitch of 440 cycles per second is the 51st note up the piano keyboard.

B. Use the same formula to find the frequency of the pitch that is 73 notes up the keyboard:

73 = 1 + 12 log₂(f/27.5)

72 = 12 log₂(f/27.5)

6 = log₂(f/27.5)

f/27.5 = 2⁶

f = 27.5 × 2⁶

f ≈ 1760

Therefore, the frequency of the pitch that is 73 notes up the keyboard is approximately 1760 cycles per second.

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Answer:

Step-by-step explanation:

You want to classify the functions shown in the tables as linear, quadratic, or exponential.

We notice all of the tables have x-values that are evenly spaced. this means we can look at the differences between y-values to determine the kind of function the table represents.

The differences have the following interpretation:

The differences of y-terms are constant at 3 -5 = -2.

The function is linear.

The differences of y-terms are constant at 5 -2 = 3.

The function is linear.

We observe that the y-values have a minimum. We don't need to take differences to know this is not linear or exponential. Of the offered choices, the only one that makes sense is "quadratic."

The differences of y-terms are ...

  1 -5 = -4, 5 -1 = 4, 17 -5 = 12, 37 -17 = 20

The differences of differences are ...

  4 -(-4) = 8, 12 -4 = 8, 20 -12 = 8

The second differences are constant.

The function is quadratic.

The first differences are ...

  3 -1 = 2, 9 -3 = 6, 27 -9 = 18, 81 -27 = 54

The second differences are

  6 -2 = 4, 18 -6 = 12, 54 -18 = 36

We note that the first and second differences are not constant, but the ratio of terms at every level is 3/1 = 6/2 = 12/4 = 3.

The function is exponential.

The missing probability is P(A or B) = 131/200

It should be noted that in order to find the probability of the union of two events A and B, i.e., P(A or B), we can use the following formula:

P(A or B) = P(A) + P(B) - P(A and B)

P(A or B) = P(A) + P(B) - P(A and B) = (9/20) + (1/4) - (9/200)

Simplifying the expression, we get:

P(A or B) = 45/100 + 25/100 - 9/200 = (90+50-9)/200 = 131/200

The probability is 131 / 200.

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Answer:

GH = 6.2

Step-by-step explanation:

∠WTA and ∠HTC are vertical angles. Thus, they are equal.

As WA || HC, that means ∠AWT and ∠THC are alternate interior angles and are equal. Additionally, ∠WAT and ∠TCH are also alternate interior angles and are equal.

∠WTA = ∠HTC, ∠AWT = ∠THC, and ∠WAT = ∠TCH

Since both triangles have the same angle measurements, they are similar.

Answer:

angle x is 160

Step-by-step explanation:

180 - 20

The matrix formed by performing the row operation 4R₁ + R₂ — R₂ on M will have R₁ = [1 0 3] and R₂ = [ -1 2 10]

Row operations are a set of operations that can be performed on the rows of a matrix in order to transform it into a row equivalent matrix, which has the same solution set as the original matrix.

performing the row operation 4R₁ + R₂ — R₂ on M, we have;

4(1) + (-5) = -1 {row 2 column 1}

4(0) + 2 = 2 {row 2 column 2}

4(3) + (-2) = 10 {row 2 column 3}

Therefore, the matrix formed by performing the row operation 4R₁ + R₂ — R₂ on M will have R₁ = [1 0 3] and R₂ = [ -1 2 10]

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The simplified expression of  2x multiplied by 7 with the exponent of 4 is 4,802x.

The expression is simplified by applying the rules of multiplication and exponent rules.

the expression = 2x(7⁴)

simplify 7⁴ as follows = 7 x 7 x 7 x 7 = 2,401

Multiply the resultant solution of 2,401 as follows;

= 2,401 x (2x)

= 4,802x

Thus, the simplified expression of  2x multiplied by 7 with the exponent of 4 is determined by applying basis rules of multiplication.

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Answer:

a) f(g(1)) = f(0) = 5

b) f(g(2)) = f(-3) = 8

c) g(f(2)) = g(3) = -8

d) g(f(3)) = g(2) = -3

e) g(g(1)) = g(0) = 2

f) f(f(3)) = f(2) = 3

Reese sold 20 candy and Skittles sold 14 candy for a total of $41. The equation for this is x + y = 34 and x + 1.5y = 41

An exponential equation is an expression that shows how numbers and variables using mathematical operators.

Let x represent the number of candy that Reese sell and y represent the number of Skittles candies

Reese's sells for $1 and skittles for $1.50

34 candies was sold, hence:

x + y = 34      (1)

He made $41, hence:

x + 1.5y = 41    (2)

Solving both equations simultaneously:

x = 20; y = 14

Reese had 20 candy and Skittles had 14 candy.

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b. The range is 32 - 16 = 16. and IQR  4

c. 32

a. Here is a box and whisker plot to represent the given data:

The horizontal axis shows the range of the data, and the vertical axis represents the frequency. The box represents the middle 50% of the data, with the bottom of the box being the 25th percentile and the top of the box being the 75th percentile. The whiskers represent the lowest and highest values within 1.5 times the interquartile range (IQR) of the box, and any points outside the whiskers are considered outliers.

b. The range is the difference between the highest and lowest values in the data set. In this case, the range is 32 - 16 = 16.

c. There is one outlier in the data set, which is the value of 32. It is outside the whiskers of the box and is considered to be an outlier.

The interquartile range (IQR) is the difference between the 75th and 25th percentiles of the data set. In this case, the IQR is:

IQR = Q3 - Q1

   = 22 - 18

   = 4

where Q1 is the first quartile (the value below which 25% of the data falls), and Q3 is the third quartile (the value below which 75% of the data falls).

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Using mathematical operations, Cindy's answer would have been 15 if she had worked the problem correctly by following her teacher's instructions.

The correct value can be determined by reversing Cindy's calculations to find the certain number as follows:

(x - 9) ÷ 3 = 43

x = 43 x 3 + 9

x = 138

With the certain number, x = 138, the correct mathematical operations can be performed as follows:

(138 - 3) ÷ 9

135 ÷ 9

= 15

Thus, Cindy's correct answer would have been 15 if she had performed the correct mathematical operations, according to her teacher's instructions.

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According to the graph, the function that best describes the graph is

[tex]f(x)=|x| = \left\{ \begin{array}{cl}x - 2 \text{ for x \lt -3 } \\8 \text{ for -3 \lt x \lt 5}\\-x + 10 \text{ for x \gt 6}\end{array} \right.[/tex]
(option d).

As we all know that the term function is defined as a rule or relationship that maps each element in one set, called the domain, to exactly one element in another set, called the range.

While we looking into the graph we have identified that there is a straight line that cross the point (0, 8) within the range of -3 to 5.

And then the next line is moves downwards from the range of x whose value is greater than 6.

Final upward slope is moves the range of -3 or less While we looking into these range value we have identified that the the function that refers the graph is (option d).

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Answer:

the average amount Mrs. Gordon pays in interest each month, we need to determine the total interest paid over the six-month period and divide it by the number of months.

The total interest paid can be found by subtracting the cost of the stove from the total amount paid over six months:

Total interest paid = Total amount paid - Cost of the stove

The total amount paid over six months is calculated by adding the monthly payments:

Total amount paid = $160/month * 6 months

Let's perform the calculations:

Total amount paid = $160/month * 6 months = $960

Total interest paid = Total amount paid - Cost of the stove

= $960 - $850

= $110

Now, to find the average amount Mrs. Gordon pays in interest each month, we divide the total interest paid by the number of months:

The average amount paid in interest each month = Total interest paid / Number of months

= $110 / 6 months

≈ $18.33

Therefore, Mrs. Gordon pays an average of approximately $18.33 in interest each month.

The calculated value of x in the right triangle is 8.7

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

The right triangle

The value of x in the right triangle can be calculated using the following sine ratio

sin(75) = x/9

Cross multiply the equation

So, we have

x = 9 * sin(75)

Evaluate the products

x = 8.7

Hence, the value of x in the right triangle is 8.7

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Where is the given proof

Answer:

  61°

Step-by-step explanation:

You want the angle XCA in the figure, given that angle XBA is 70°, and ABCD is a rectangle with AB=5.6m and BC=6.4m. Angles at A are right angles.

The length of diagonal AC is found using the Pythagorean theorem.

  AC² = AB² +BC²

  AC = √(5.6² +6.4²) = 0.8√113 ≈ 8.504 . . . . meters

The length XA is found using the tangent function:

  Tan = Opposite/Adjacent

  tan(B) = XA/AB

  XA = AB·tan(B) = 5.6·tan(70°) . . . . meters

The angle XCA is found from the tangent relation:

  tan(XCA) = XA/AC

  tan(XCA) = 5.6·tan(70°)/(0.8√113) = 7·tan(70°)/√113

  angle XCA = arctan(7·tan(70°)/√113) ≈ 61.07°

The angle XC makes with the horizontal is about 61°.

Step-by-step explanation:

V = 1/3 x π x r² x h

48 = 1/3 x π x r² x 9

48÷3π = r²

√48÷3π =r

so, r= 2.26 (3.s.f)

It will take Lisa and Felipe approximately 4.95 hours to fill the order by working together.

To solve this problem, we can use the formula:

1/T = 1/T₁ + 1/T₂

where T is the time taken to complete the job when both Lisa and Felipe work together, T₁ is the time taken by Lisa to complete the job, and T₂ is the time taken by Felipe to complete the job.

Substituting the given values into the formula, we get:

1/T = 1/9 + 1/11

Simplifying the right-hand side of the equation, we get:

1/T = (11 + 9)/(9 x 11)

1/T = 20/99

Multiplying both sides by 99, we get:

T = 99/20

Therefore, working together, Lisa and Felipe can complete the job in 4.95 hours, or approximately 4 hours and 57 minutes.

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The whisker plot with the data provided should includes a range that goes from 7 to 49, and a median of 23.

A whisker plot, also known as a box plot, is a graphical representation of a data set that shows the distribution of the data along with some important statistical information, such as the median, quartiles, and outliers. It is particularly useful for comparing the distribution of different data sets or for identifying potential outliers in a data set.

To create a whisker plot, follow these steps:

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The proportion is solved and ratio of minutes spent talking to teachers to minutes spent talking to students is given by 2 : 1

Given data ,

The principal spends 96 minutes talking to the teachers

And , the principal spends 48 minutes talking to students

Now , the basic ratio of minutes spent talking to teachers to minutes spent talking to students, we need to simplify the ratio by dividing both values by their greatest common factor

So , A : B = 96 : 48

On simplifying the proportion , we get

A : B = 2 : 1

Hence , the proportion is 2 : 1

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The practical domain of the situation is [0, ∞].

The practical range of the situation is [0, 100].

In Mathematics and Geometry, a domain refers to the set of all real numbers (x-values) for which a particular function (equation) is defined.

In Mathematics and Geometry, the horizontal portion of any graph is used to represent all domain values and they are both read and written from smaller to larger numerical values, which simply means from the left of any graph to the right.

By critically observing the graph shown in the image attached above, we can reasonably and logically deduce the following domain and range:

Domain = [0, ∞] or 0 ≤ x ≤ ∞.

Range = [0, 100] or 0 ≤ y ≤ 100.

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The ratio of the surface area of Pyramid A to Pyramid B is: 36/49

We have the information from the question is:

Pyramid A and Pyramid B are similar.

The volume of Pyramid A is 648 [tex]m^3[/tex]

The volume of Pyramid B is 1029 [tex]m^3[/tex]

To find the ratio of the surface areas of Pyramid A to Pyramid B.

Now, As we know that:

If two solids are similar, then the ratio  of their volumes is equal to the cube

of the ratio of their corresponding linear measures.

[tex]\frac{Vol of A}{Vol of B} =(\frac{a}{b} )^3 =\frac{684}{1029}=\frac{216}{343}\\ \\ \frac{a}{b}=\frac{6}{7}[/tex]

If two solids are similar, then the n ratio of their surface areas is equal  to the square of the ratio of their  corresponding linear measures.

Surface area of A/ Surface area of B [tex]=(\frac{a}{b} )^2=\frac{36}{49}[/tex]

So, the ratio of the surface area of Pyramid A to Pyramid B is: 36/49

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Answer:

1.235 kg

Step-by-step explanation:

Let's assume that each packet of flour and salt weighs the same.

When we remove 7 packets of flour, we are left with 5 packets of flour and 8 packets of salt. The total number of packets is 13 and their total mass is 9.9 kg.

Therefore, the average mass of each packet is 9.9 kg ÷ 13 = 0.76 kg.

Since we know that 12 packets of flour and 8 packets of salt have a total mass of 19 kg, we can subtract the mass of the 12 packets of flour (12 × 0.76 kg = 9.12 kg) from the total mass to find the mass of the 8 packets of salt:

19 kg - 9.12 kg = 9.88 kg

Finally, we can divide the mass of the 8 packets of salt by the number of packets to find the mass of each packet:

9.88 kg ÷ 8 = 1.235 kg

Therefore, each packet of salt weighs approximately 1.235 kg.

Answer: 5.3 repeated portions or about 5 portions

Step-by-step explanation: 25 pounds = 400 ounces

400/75 = 5.3 repeated

Answer:33

Step-by-step explanation:

Note that 7th grade mean = 277.86

the 5th grade mean = 254.77

So th relationship between the mean is that the 7 th grade mean is higher than the 5 th grade mean .

To compute the means here   is what we did

7 th  grade mean = (1 × 10) + (1 × 11) +   (2 × 12) + (1 × 13) +   (1 × 14) + (2 × 15) + (3 × 16) + (3 × 17) + (2 × 18) + (2 × 19) +   (3 × 20) / 21

= 277.857142857

≈ 277.86

For the 5th grade mean

5th  grade mean = (1 × 8)   + (2 × 9) + (2 × 10) +   (2 × 11) + (2 × 12) + (1 × 13) + (3 × 15) + (2 ×16) + (1 × 17)  +  (2 × 18) + (1 × 19) + (1   × 20) / 26 = 12.5

= 254.769230769

≈ 254.77

This means that  trully,  the relationship between the mean is that the 7  th grade mean is higher than the 5 th grade mean.

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The equation in slope-intercept form for the line that has slope of -2/5 and y-intercept of -4 is written as: y = -2/5x - 4.

The slope-intercept form for any given straight line or linear relationship can be expressed as y = mx + b, where:

m = slope of the line

b = the y-intercept

Given the following:

slope (m) = -2/5.

y-intercept (b) = -4

To write the equation, substitute m = -2/5 and b = -4 into y = mx + b:

y = -2/5x - 4

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The statement that is true about the given circle equation is:

Option A: The circle is centered at (−6, 3) and has a radius of 11.

The standard form for finding the equation of a circle is given by the expression:

(x - h)² + (y - k)² = r²

where:

(h, k) represents the coordinates of the center of circle

r represents the radius of the circle

We are given the equation of the circle as:

(x - 6)² + (y - 3)² = 121

Now, 121 can also be written as 11². Thus, the equation of the circle can be rewritten as:

(x - 6)² + (y - 3)² = 11²

Comparing with the standard form of equation of a circle gives:

Coordinates of center = (-6, 3)

Radius = 11

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Answer:

1. True

2.True

3.True

4.False

5.False

6.True

7.True

8.True

Step-by-step explanation:

1. The area of a rectangle is found by the formula A=bh.

2. There are 4 rectangles making up the lateral area of the prism. 2+3=5. 5*6=30, and multiply by 2 since there are two of the 2x6 and two of the 3x6 rectangles to get 60. The bases of the prism have an area of 6 square inches each.

60+12+12=84

3. 3*5=15

4. The base is not a square, so you cannot multiply by 4 to get the area of all the triangles. You can multiply by two, and also multiply 1/2(3)(4) by 2 to get the area of the other two triangles.

5. The base is a triangle. 1/2(3)(4)=6.

6. 2[1/2(3)(4)]=12. 2(8*5)=80. 3*8=24. 80+24+12=116

7. A cube has 6 identical sides, so you can find the area of one side and multiply by 6 to get the surface area of the cube.

8. 9^2=81. 81*6=486.