An independent samples t-test would be appropriate to analyze the difference in running speed between rats that received a steroid drug supplement and those that did not.
The appropriate statistical test to analyze the difference in running speed between two groups of rats (one receiving a steroid drug supplement and one not receiving it) would be a t-test. Specifically, an independent samples t-test would be suitable for this type of comparison, assuming that the data meets the assumptions for parametric testing.
This test would compare the means of the two groups and assess the probability that the observed difference is due to chance alone. A significant result would indicate that the difference in running speed is unlikely to be due to chance and is instead likely to be a true effect of the steroid drug supplement
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what determines the decision of whether to use a one-tailed test or a two-tailed test when testing hypotheses to make inferences about a population mean?
The decision to use a one-tailed or two-tailed test depends on the specific research question and the directionality of the hypothesis.
When testing a hypothesis about a population mean, researchers may use a one-tailed or two-tailed test. A one-tailed test is used when the research question only focuses on one direction of an effect, for example, if the researcher is interested in whether the population mean is greater than a specific value. In contrast, a two-tailed test is used when the research question is interested in any significant difference between the sample mean and the hypothesized population mean, regardless of the direction of the difference.
The decision to use a one-tailed or two-tailed test should be based on the specific research question, and it is essential to clearly define the directionality of the hypothesis before selecting the appropriate test.
Therefore, the choice of a one-tailed or two-tailed test should be based on the research question and the specific direction of the hypothesis.
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Can anyone help me out on this please?
Simplifying rational expressions.
Answer:
A
Step-by-step explanation:
Cost for x shirts in a week (4x+150) divided by the shirts made in a week (x)
would be represented by A
Answer:
A is correct.
Step-by-step explanation:
The expression 4x + 150 represents the cost of manufacturing x shirts in a week. If x shirts are made in a week, the expression [tex]\frac{4x+150}{x}[/tex] (Answer A) represents the manufacturing cost per t-shirt (that's why we divide by x) where x shirts are made in a week (that's why we divide by just x, not 7x, because we're talking about 1 week, not 7 days)!
Finding the area of a triangle is straightforward if you know the length of the base and the height of the triangle. But is it possible to find the area of a triangle if you know only the coordinates of its vertices? In this task, you’ll find out. Consider ΔABC, whose vertices are A(2, 1), B(3, 3), and C(1, 6); let line segment AC represent the base of the triangle.
a. Find the equation of the line passing through B and perpendicular to AC
Type your response here:
b. Let the point of intersection of AC with the line you found in part a be point D. Find the coordinates of point D.
Type your response here:
c. Use the distance formula to find the length of the base and the height of ABC
Type your response here:
d. Find the area of ABC
Type your response here:
The line that is parallel to AC and passes through B has the equation y: 1/5x + 12/5.
What is the equation of the line?The formula for a straight line is y=mx+c where c is the height at which the line intersects the y-axis, also known as the y-intercept, and m is the gradient.
The benefit of expressing a linear equation in this way is that the values for m and b correspond to the slope and y-intercept, respectively.
This makes it easy to graph the line that the equation describes.
So, the slope of AC is:
6-1/1/2= 5/-1 = -5
The equation of line D, which passes through B and is perpendicular to AC, is given as y: mx + p.
Where p is the y value of the y-intercept point and m is the slope.
The fact that line D is perpendicular to line AC indicates that their slopes add up to one.
Therefore,
(−5) × m = −1
Now,
m = -1/-5 = 1/5
We discover that the D equation is:
Y: 1/5x + p
To complete the equation of D at this time, we still require the value of p.
B(3, 3)∈ D
3 = 1/5(3) + p
p = 3 - 3/5 = 12/5 = 2.4
Hence, equation D will be:
y: 1/5x + 12/5
Therefore, the line that is parallel to AC and passes through B has the equation y: 1/5x + 12/5.
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Correct question:
Finding the area of a triangle is straightforward if you know the length of the base and the height of the triangle. But is it possible to find the area of a triangle if you know only the coordinates of its vertices? In this task, you’ll find out. Consider ΔABC, whose vertices are A(2, 1), B(3, 3), and C(1, 6); let line segment AC represent the base of the triangle.
Part A
Find the equation of the line passing through B and perpendicular to AC .
Find the surface area of the figure. Round to the nearest hundredth when necessary (2 decimal places). SA = hp + 2B
The calculated surface area of the figure is 3100.8 square feet
Calculating the surface area of the figurefrom the question, we have the following parameters that can be used in our computation:
SA = hp + 2B
From the figure, we have
B = 1/2 * (10 + 34) * 24.7 = 543.4
p = 2 * (34 + 19) = 106
h = 19
So, we have
SA = 19 * 106 + 2 * 543.4
Evaluate
SA = 3100.8
Hence, the surface area is 3100.8 square feet
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Syed is comparing train distances between his hometown and other nearby towns with the cost of travel fares. His results are shown in the table below. He has modeled the data with the equation y=0.1x + 12
Using the sketchpad, calculate and record the missing residuals so that Syed can determine whether a linear model is appropriate for his data set.
The data set as there are some large residuals, such as 4 and -7, indicating that the actual fares are not well-predicted by the linear equation.
Define the term residuals?In statistics, residuals refer to the differences between the observed values of a variable and the values predicted by a statistical model. They are calculated by subtracting the predicted values from the observed values.
To calculate the residuals, we first need to find the predicted fares for each distance using the equation y = 0.1 x + 12;
Miles Fares $ Predicted Fares $ Residuals $
50 14 17 -3
40 16 16 0
60 16 18 -2
30 14 15 -1
60 22 18 4
70 12 19 -7
80 24 20 4
To find the residuals, we subtract the predicted fare from the actual fare. If the predicted fare is greater than the actual fare, the residual is negative, and if the predicted fare is less than the actual fare, the residual is positive. We can see that a linear model may not be appropriate for the data set as there are some large residuals, such as 4 and -7, indicating that the actual fares are not well-predicted by the linear equation.
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In a rectangle, the length is 5m less than twice the width and the area is 97m^2 . Approximate the dimensions to the nearest tenth of a meter. Do not round any intermediate steps. Find the length and width.
Therefore, the approximate dimensions of the rectangle to the nearest tenth of a meter are width = 7.9 m and length = 10.8 m.
What is area?Area is a measure of the size of a two-dimensional surface or shape. It is typically expressed in square units, such as square meters (m²), square centimeters (cm²), square feet (ft²), or square inches (in²).
Here,
Let's denote the width of the rectangle as "w" in meters. Then, according to the problem statement, the length of the rectangle can be expressed in terms of the width as:
length = 2w - 5
The area of the rectangle is given as 97 square meters, so we can write:
area = length × width
Substituting the expression for length in terms of the width, we get:
area = (2w - 5) × w
Expanding this expression, we get a quadratic equation in standard form:
2w² - 5w - 97 = 0
We can solve this equation using the quadratic formula:
w = (-(-5) ± √((-5)² - 4×2×(-97))) / (2×2)
Simplifying this expression, we get:
w ≈ 7.9 or w ≈ -6.2
Since the width of a rectangle cannot be negative, we discard the negative solution and approximate the width to the nearest tenth of a meter:
w ≈ 7.9 m
Using the expression for the length in terms of the width, we can find the length:
length = 2w - 5
≈ 10.8 m
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Order the set of numbers from least to greatest: (4 points)
2.71 repeating 71, 2 and 3 over 4, square root 5, 5 over 2
Group of answer choices
2.71 repeating 71, 2 and 3 over 4, square root 5, 5 over 2.
square root 5, 5 over 2, 2.71 repeating 71, 2 3 over 4
5 over 2, square root 5, 2.71 repeating 71, 2 3 over 4
2 3 over 4, 2.71 repeating 71, 5 over 2, square root 5
The correct order of the set of numbers from least to greatest is 2, 3 over 4, 2.71 repeating 71, 5 over 2, square root 5 option (D) is correct.
To order the given set of numbers from least to greatest, we need to compare their values. We start by noticing that 2 and 3 over 4 are equivalent to 2.75. Next, we compare the values of the remaining numbers. Since the square root of 5 is approximately 2.236 and 5 over 2 is equivalent to 2.5, we have:
2, 3 over 4 < 2.71 repeating 71 < 2.5 < square root 5
Therefore, the set of numbers in order from least to greatest is 2, 3 over 4, 2.71 repeating 71, 5 over 2, square root 5.
Ordering the set of numbers from least to greatest, we get:
2, 3 over 4, 2.71 repeating 71, square root 5, 5 over 2.
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The complete question is:
Order the set of numbers from least to greatest: (Group of answer choices)
A. 2.71 repeating 71, 2, and 3 over 4, square root 5, 5 over 2.
B. square root 5, 5 over 2, 2.71 repeating 71, 2 3 over 4
C. 5 over 2, square root 5, 2.71 repeating 71, 2 3 over 4
D. 2, 3 over 4, 2.71 repeating 71, 5 over 2, square root 5
In its simplest form work out 5/14 x 7/30
Answer:
[tex]\boxed{\frac{1}{12}}[/tex]
Step-by-step explanation:
The multiplication of fractions satisfies the following rule:
[tex]\displaystyle{\frac{a}{b}\times\frac{c}{d}=\frac{a\times c}{b\times d}[/tex]
applied to our exercise, it looks like this
[tex]\displaystyle{\frac{5}{14}\times\frac{7}{30}=\frac{5\times 7}{14\times 30}[/tex]
also, we can "decompose" denominator numbers as follows:
[tex]14=7*2[/tex]
[tex]30=5*6[/tex]
Therefore:
[tex]\displaystyle\frac{\not{5}\times \not{7}}{2 \times \not{7}\times \not{5}\times 6}= \frac{1}{12}[/tex]
In this way, only one multiplication was necessary.
[tex]\text{-B$\mathfrak{randon}$VN}[/tex]
What is the total area, in square inches, of the shaded sections of the trapezoid below. ?
Total area of shaded region in trapezoid is 54.4in²
Define trapezoidA trapezoid is a four-sided, quadrilateral shape with two parallel sides, which are called the bases, and two non-parallel sides, which are called the legs.
The formula for the area of a trapezoid is:
Area = (1/2) × (sum of the bases) × (height)
The Shaded region in trapezoid form right angled triangle.
Area of 1st shaded region=area of right triangle
=1/2×b×h
=1/2×6.8×7.4
=25.16in²
Area of 2nd shaded region=area of right triangle
=1/2×b×h
=1/2×6.8×8.6
=29.24in²
Hence, total area of shaded region in trapezoid is 54.4in²
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On a map, the distance between Dallas Texas and Memphis Tennessee is 57.5 cm what is the actual distance if the map scale 1 cm:8mi
Answer:
57.5 cm × 8 mi./cm = 460 miles
The circle graph shows how a family spends its annual Income. If $25,200 is used for Food, Savings, and Insurance combined, what is the total annual income?
housing 21%
food 18%
clothing 17%
auto 14%
entertaining 12%
insurance 11%
savings 7%
Answer:41,328
Step-by-step explanation:
18+7+11=36 36%
100-36=64 64%
25,200 x 164%=41,328
For the following problems, are they similar? if so why? (AA~, SAS~, SSS~)
Answer:
yes these 2 triangles are similar
Step-by-step explanation:
triangle ABC is similar to triangle ADE. it has a scale factor of 1.5. It is a dilated. 10×1.5=15 so it has a scale factor of 1.5 which is an enlargement. any scale factor greater than 1 increases the size of the triangle. it is proportional if you multiply both sides by the scale factor (1.5) you get 15. also The angles are equal (AA similarity) these two triangles are NOT congruent but they are similar the degrees are equal. btw SSS and SAS are congruence postulates NOT similarity. so therefore they are similar because of AA
i hope this helps :)
Carmen is trying to choose between two bike rental companies where h
is the number of hours rented and y
is the cost, in dollars. Company A charges a $30
fee and $2
for each hour rented, which can be represented by the equation y=2h + 30
. Company B charges a $10
fee and $4
for each hour rented, which can be represented by the equation y = 4h + 10
.
Select the coordinate below that represents the cost when renting from either company would be the same.
(10, 50)
(30, 50)
(50, 10)
(50, 30)
the coordinate that represents the cost when renting from either company would be the same is (10, 50). This means that if Carmen rents a bike for 10 hours, she would pay the same amount whether she rented from Company A or Company B.
How to determine the point?
To determine the point where the cost of renting a bike from Company A and Company B is the same, we need to find the values of h and y that satisfy the equations y = 2h + 30 and y = 4h + 10. We can solve for h by setting the two equations equal to each other:
2h + 30 = 4h + 10
Simplifying this equation, we get:
20 = 2h
h = 10
Now that we know h is 10, we can substitute that value into either equation to find y. Using the first equation:
y = 2h + 30 = 2(10) + 30 = 50
Therefore, the coordinate that represents the cost when renting from either company would be the same is (10, 50). This means that if Carmen rents a bike for 10 hours, she would pay the same amount whether she rented from Company A or Company B.
It's worth noting that as the number of hours rented increases, the difference in cost between the two companies becomes greater. For example, if Carmen rented a bike for 20 hours, the cost would be $70 from Company A and $90 from Company B. Therefore, it's important for Carmen to consider how long she plans to rent the bike for when deciding which company to rent from.
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Bootle Corp paid Leslie a quarterly dividend payment for $1,490. Leslie
owns 119 shares of Bootle. What was the quarterly dividend for one share
of Bootle?
The quarterly dividend fοr οne share οf Bοοtle is $12.52.
What is divisiοn?A divisiοn is οne οf the fundamental mathematical οperatiοns that divides a larger number intο smaller grοups with the same number οf cοmpοnents. Hοw many tοtal grοups will be established, fοr instance, if 20 students need tο be separated intο grοups οf five fοr a spοrting event?
The divisiοn οperatiοn makes it simple tο tackle such issues. Divide 20 by 5 in this case. 20 x 5 = 4 will be the οutcοme. There will therefοre be 4 grοups with 5 students each. By multiplying 4 by 5 and receiving the result 20, yοu may cοnfirm this value.
Tο find the quarterly dividend fοr οne share οf Bοοtle, we can divide the tοtal quarterly dividend payment by the number οf shares Leslie οwns:
Quarterly dividend per share = Tοtal quarterly dividend payment / Number οf shares
Quarterly dividend per share = $1,490 / 119
Quarterly dividend per share = $12.52 (rοunded tο twο decimal places)
Therefore, the quarterly dividend for one share of Bootle is $12.52.
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acellus find the surface area of the composite figure its the yellew and green one the green is the triangle and the yellow is the rectangle
The surface area of the composite figure is 446 cm².
What is surface area?Surface area is the sum of the areas of all faces (or surfaces) on a 3D object. It is a two-dimensional measure of the space occupied by the object, and is measured in units of square units, such as square inches, square feet, and square meters. Surface area is often used to determine the amount of material needed for a solid object or the amount of paint needed to cover it.
The surface area of the composite figure can be calculated using the formula SA = 2(L x W) + (2 x H x L) + (2 x H x W). In this case, L = 9 cm, W = 10 cm, and H = 7 cm.
Therefore, the surface area of the composite figure is SA = 2(9 cm x 10 cm) + (2 x 7 cm x 9 cm) + (2 x 7 cm x 10 cm) = 180 cm²+ 126 cm²+ 140 [tex]cm^2[/tex] = 446 cm².
Therefore, the surface area of the composite figure is 446 [tex]cm^{2}[/tex].
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Complete questions as follows-
Surface Area of Composite Figures Acellus Find the surface area of the composite figure. 1 9 cm 10 cm cm 7 cm 8 cm 2 cm SA [? ] cm? Enter Copyright 02003 2021 Acellus Corporation. All Rights Reserved:'
Find the area of the figure below in square units.
Answer: 48.5 sq units
Step-by-step explanation:
7*8=56, 56-7.5= 48.5
Help, please I need it now
Step-by-step explanation:
Find the frontal area = triangle PLUS square...then multiply by depth ,a.
1/2 * a * b + a * a all of this times the depth 'a'
[ 1/2 ( 7 *5) + 7*7 ] * 7 = 465.5 units^3
Alg 1 - 250 toothpick pyramid task
Write a function f(l) that determines the number of triangles in any given level of the pyramid.
f(l) =
pls help asapppp
Answer: The answer is 7
Step-by-step explanation:
there are 23 23 guests at your home for a dinner party. in how many ways could the guests arrange themselves on a four person couch?
There are 212520 ways by which guests arrange themselves on a four person coach.
What is Permutation?The term permutation refers to a mathematical calculation of the number of ways a particular set can be arranged. Put simply, a permutation is a word that describes the number of ways things can be ordered or arranged. With permutations, the order of the arrangement matters.
there are total 23 guests and they arranged themselves in 4 person coach,
so,
n = 23 and r = 4
npr = n!/(n-r)!
= 23!/(23-4)!
= 23!/19!
= 23x 22 x 21 x 20x 19!/19!
now cancel the factorial 19 from numerator as well as denominator
= 23x 22 x 21 x 20
= 212520
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Please provide a through explanation its much appreciated :)))
Part A: The side length of triangle EF = 10.35.
Part B: ∠E = 78
Define about the sine rule:The law of sines, often known as the sine rule or sine formula, is a formula that links a triangle's sides to the sine of its diagonally opposed angles.Both the sides and the angles of a triangle can be determined using the law of sine.You must apply the form of the sine rule at which lengths are now the numerators in order to determine the length of a side: a/Sine (A) = b/Sine (B) = c/Sine (C)Sine rule :
e/sinE = f/sinF = d/sinD
Where D, E, and F are the angles and d,e and f are the angles' opposing sides.
Put the values:
DF/sinE = 11.7/sin55 = EF/sin47
Part A: EF = ?
11.7/sin55 = EF/sin47
EF = 11.7 * sin47 / sin55
EF = 10.35
Part B: ∠E = ?
Using angle sum property of triangles.
∠E + ∠D + ∠F = 180
∠E + 47 + 55 = 180
∠E = 78
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Given sinx=725 and cosx=2425. What is ratio for tanx ?
The ratio for tanx is 7/24.
To find the ratio for tanx, we need to use the given values of sinx and cosx.
The formula for tanx is:
tanx = sinx / cosx
Given, sinx = 7/25 and cosx = 24/25.
Now, let's substitute these values in the formula:
tanx = (7/25) / (24/25)
To divide these fractions, multiply the first fraction by the reciprocal of the second fraction:
tanx = (7/25) * (25/24)
Now, simplify the fractions:
tanx = (7 * 25) / (25 * 24)
The 25 in the numerator and denominator will cancel out:
tanx = 7 / 24.
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5. Chang says the longest ski trail is more than three times the length of the shortest ski trail.
Is Chang correct? Explain.
The statement of Chang's is not correct.
Here given Chang's equation is x = 3y, which means that the length of the longest ski trail (x) is three times the length of the shortest ski trail (y).
Therefore, if we assume that the length of the shortest ski trail is y, then the length of the longest ski trail would be 3y.
So, in Chang's statement, he says that the longest ski trail is more than three times the length of the shortest ski trail. This would mean that x > 3y. However, according to Chang's equation, x = 3y, which contradicts his statement.
Therefore, Chang is incorrect in saying that the longest ski trail is more than three times the length of the shortest ski trail. In fact, the longest ski trail is exactly three times the length of the shortest ski trail, as per Chang's equation.
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Correct question is "Chang has an equation x=3y where x is length of longest ski trail and y is length of the shortest ski trail.
Chang says the longest ski trail is more than three times the length of the shortest ski trail.
Is Chang correct? Explain."
5x−5≥− 18 what is x ?
Answer:
5x−5≥−185x-5≥-18Move all terms not containing xx to the right side of the inequality.Tap for more steps...5x≥−135x≥-13Divide each term in 5x≥−135x≥-13 by 55 and simplify.Tap for more steps...x≥−135x≥-135The result can be shown in multiple forms.Inequality Form:x≥−135x≥-135Interval Notation:[−135,∞)
The required answer is x represents any real number greater than or equal to -13/5 in terms of the result of solving linear inequalities,
Given that, linear inequalities 5x−5≥− 18
To solve the inequality 5x - 5 ≥ -18, we can follow these steps:
Step 1: Add 5 to both sides of the inequality to isolate the term with x:
5x - 5 + 5 ≥ -18 + 5
This simplifies to:
5x ≥ -13
Step 2: Divide both sides of the inequality by 5 to solve for x:
(5x)/5 ≥ (-13)/5
Simplifying further:
x ≥ -13/5
So, the solution to the solving linear inequality is x ≥ -13/5.
In terms of the result of solving linear inequalities, x represents any real number greater than or equal to -13/5.
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Mia is building a fence to form a triangle shaped garden. She has built one side 24 ft. long and another side 15 ft. long. What length could Mia build the last side of the fence? Responses 7 ft 7 ft 8 ft 8 ft 10 ft 10 ft 9 ft
To determine the length of the third side of the triangle, we can use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In this case, we have two sides of lengths 24 ft and 15 ft. Let x be the length of the third side. Then, according to the triangle inequality theorem, we have:
24 + 15 > x
39 > x
and
15 + x > 24
x > 9
Combining these inequalities, we have:
9 < x < 39
Therefore, the length of the third side of the fence could be any value between 9 ft and 39 ft, but it cannot be exactly 7 ft, 8 ft, or 10 ft. Out of the given options, the closest value to the range is 9 ft.
suppose that a random sample of size 36 is to be selected from a population with mean 44 and standard deviation 8. what is the approximate probability that x will be more than .5 away from the population mean?
There is a 75.40% chance that the sample mean X will be more than 0.5 away from the population mean μ.
Let X be the random variable representing the sample mean of a random sample of size 36 selected from a population with mean μ=44 and standard deviation σ=8.
The Central Limit Theorem (CLT) states that, under certain conditions, the distribution of the sample mean X can be approximated by a normal distribution with mean μ and standard deviation σ/√(n), where n is the sample size. Since the sample size is 36, we can use this approximation.
The probability that X is more than 0.5 away from the population mean can be calculated as follows:
P(|X-μ| > 0.5) = P((X-μ)/ (σ/√(n)) > (0.5) / (σ/√(n))) + P((X-μ)/ (σ/√(n)) < (-0.5) / (σ/√(n)))
Using the standard normal distribution table, we can find the corresponding probabilities for each term on the right-hand side of the equation. We have:
P(Z > 0.3125) + P(Z < -0.3125)
where Z is a standard normal random variable with mean 0 and standard deviation 1.
Using the standard normal distribution table, we can find that P(Z > 0.3125) = 0.3770 and P(Z < -0.3125) = 0.3770.
Therefore, the approximate probability that X will be more than 0.5 away from the population mean is:
P(|X-μ| > 0.5) = 0.3770 + 0.3770 = 0.7540
In terms of percentage, we can rewrite it as,
P(|X-μ| > 0.5) = 0.7540*100 = 75.40%
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What is the value of 1/6+3/6+1/12
A. 5/12
B. 7/12
C. 9/12
D. 8/12
Answer: First, we can simplify 1/6 and 3/6 by finding a common denominator of 12:
1/6 = 2/12
3/6 = 6/12
Now we have:
2/12 + 6/12 + 1/12
Combining the numerators, we get:
9/12
Simplifying this fraction by dividing the numerator and denominator by their greatest common factor of 3, we get:
3/4
Therefore, the answer is C. 9/12.
Divide a line segment of 140cm into the proportion 2:3; 5 How long is the shortest piece!
Answer:
28cm
Step-by-step explanation:
see image for explanation.
What is the surface area of a rectangular prism with a length of 5 cm, width of 6 cm, and a height of 7 cm?
The surface area of a rectangular prism can be found using the formula:
A = 2lw + 2lh + 2wh
where l, w, and h are the length, width, and height of the prism, respectively.
Substituting the given values, we get:
A = 2(5 cm)(6 cm) + 2(5 cm)(7 cm) + 2(6 cm)(7 cm)
A = 60 cm² + 70 cm² + 84 cm²
A = 214 cm²
Therefore, the surface area of the rectangular prism is 214 cm².
chris l please show me how to solve this question: there are 10 questions on a discrete structures final exam. how many ways are there to assign scores to the problems if the sum of the scores is 100 and each question is worth at least 5 points?
There are 2,139,792,767,488,000 ways to assign scores to the problems if the sum of the scores is 100 and each question is worth at least 5 points.
To solve this problem, we will use the concept of combinations with repetition.
First, let's subtract 5 points from each question, since each question is worth at least 5 points. Now, we need to distribute the remaining 50 points (100 - 10*5) among the 10 questions.
Imagine that we have 50 identical balls (points) to distribute among 10 distinct boxes (questions). To do this, we can use a technique called "stars and bars". We need to place 9 dividers (bars) between the 50 balls (stars) to divide them among the 10 questions.
We will have a total of 59 positions (50 stars + 9 bars). We need to choose 9 of these positions for the bars, and the stars will fill the remaining positions.
The number of ways to do this is given by the combination formula:
C(n, k) = n! / (k! * (n - k)!)
In our case, n = 59 (total positions) and k = 9 (dividers). So, we need to calculate C(59, 9):
C(59, 9) = 59! / (9! * (59 - 9)!)
C(59, 9) = 59! / (9! * 50!)
C(59, 9) = 2,139,792,767,488,000
Therefore, there are 2,139,792,767,488,000 ways to assign scores to the problems if the sum of the scores is 100 and each question is worth at least 5 points.
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) let a1 be the event that the first marble is red and let a2 be the event that the second marble is red. are a1 and a2 independent? cs 70, spring 2023, hw 09 2 (b) what is the probability that rachel wins the game? (c) given that rachel wins the game, what is the probability that all of the marbles were red?
a) A and B are independent events.
b) The outcome of each game is independent of the other games so, Rachel's loss = p.
c) The probability of all marbles being red is 4
a) Given that a first marble is red and the second marble is also red, such that
P(1) = P(2) = 0.2, P(3) = P(5) = P(6) = 0.1 and P(4) = 0.3 and die is thrown two times. Also given that:
A1 = red
B1 = red.
So, P(A) = [P(1, 1) + P(2, 2) + P(3, 3) + P(4, 4) + P(5, 5) + P(6, 6)]
= P(1).P(1) + P(2).P(2) + P(3).P(3) + P(4).P(4) + P(5).P(5) + P(6).P(6)
= 0.2 x 0.2 + 0.2 x 0.2 + 0.1 x 0.1 + 0.3 x 0.3 + 0.1 x 0.1 + 0.1 x 0.1
= 0.04 + 0.04 + 0.01 + 0.09 + 0.01 + 0.01 = 0.20
Now, B = [(4, 6), (6, 4), (5, 5), (5, 6), (6, 5), (6, 6)]
P(B) = [P(4).P(6) + P(6).P(4) + P(5).P(5) + P(5).P(6) + P(6).P(5) + P(6).P(6)]
= 0.3 x 0.1 + 0.1 x 0.3 + 0.1 x 0.1 + 0.1 x 0.1 + 0.1 x 0.1 + 0.1 x 0.1
= 0.03 + 0.03 + 0.01 + 0.01 + 0.01 + 0.01 = 0.10
A and B both events will be independent if
P(A ⋂ B) = P(A).P(B) …. (i)
And, here (A ⋂ B) = {(5, 5), (6, 6)}
So, P(A ⋂ B) = P(5, 5) + P(6, 6) = P(5).P(5) + P(6).P(6)
= 0.1 x 0.1 + 0.1 x 0.1 = 0.02
From equation (i) we get,
0.02 = 0.20 x 0.10
0.02 = 0.02
Therefore, we can say that A and B are independent events.
b) We know that the outcome of each game is independent of the other games. Rachel's loss = p. As per the given information, the games in the series are unrelated, indicating the application of the probability multiplication principle.
c) The probability of all marbles being red is 4
Let a be the probability that the first player (ultimately) wins if the two players are tied in wins. Let b be the probability that she wins if she is 1 ahead. And let c be the probability she wins if she is 1 behind. We have the equations
a = rb + 1 - r x c
b = r + 1 - r x a
c = r x a
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