There exists at least one c in the open interval (a, b) such that f'(c) = 0.
There are 3 distinct points where p'(x) = 0, which corresponds to the 3 distinct real solutions of the given equation.
To prove that the given equation has exactly 3 distinct real solutions, let's follow the steps mentioned in the question.
First, consider the polynomial p(x) = (x-a)(x-b)(x-c)(x-d). Since a, b, c, and d are distinct real numbers, p(x) has 4 distinct real roots, namely a, b, c, and d.
Now, let's find the derivative p'(x) using logarithmic differentiation. Taking the natural logarithm of both sides, we have:
[tex]ln(p(x)) = ln((x-a)(x-b)(x-c)(x-d))[/tex]
Differentiating both sides with respect to x, we get:
[tex]p'(x)/p(x) = 1/(x-a) + 1/(x-b) + 1/(x-c) + 1/(x-d)[/tex]
Multiplying both sides by p(x) and simplifying, we have:
[tex]p'(x) = (x-b)(x-c)(x-d) + (x-a)(x-c)(x-d) + (x-a)(x-b)(x-d) + (x-a)(x-b)(x-c)[/tex]
Now, we apply Rolle's Theorem, which states that if a function is continuous on the closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one c in the open interval (a, b) such that f'(c) = 0.
Since p(x) has 4 distinct real roots, there must be 3 intervals between these roots where the function p(x) satisfies the conditions of Rolle's Theorem. Therefore, there are 3 distinct points where p'(x) = 0, which corresponds to the 3 distinct real solutions of the given equation.
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Find the mass of the thin bar with the given density function p(x) = 2 + x^2; for 0 ≤ x ≤ 1 О 0 O 7/3
O-2 O 2
The mass of the thin bar with the given density function is 7/3.
To find the mass of the thin bar with the given density function p(x) = 2 + x^2 for 0 ≤ x ≤ 1:
You need to integrate the density function over the given interval.
Here are the steps to do that:
STEP 1: Set up the integral for mass:
Mass = ∫(density function) dx from x = 0 to x = 1
STEP 2:Plug in the given density function:
Mass = ∫(2 + x^2) dx from x = 0 to x = 1
STEP 3:Integrate the function with respect to x:
Mass = [2x + (x^3)/3] evaluated from x = 0 to x = 1
STEP 4: Evaluate the integral at the limits:
Mass = (2(1) + (1^3)/3) - (2(0) + (0^3)/3)
STEP 5: Simplify the expression:
Mass = (2 + 1/3) - 0
STEP 6: Calculate the final mass:
Mass = 2 + 1/3 = 7/3
So, the mass of the thin bar with the given density function is 7/3.\
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A recipe for snack mix has a ratio of 2 cups nuts, 4 cups pretzels, and 3 cups raisins. How many cups of nuts are there for each cup of raisins?
Answer: 1 cups of nuts : 1 1/2 cups of raisins
Step-by-step explanation:
The student council wants to determine the best date for the end-of-year dance for the 350 students. They survey every 5th student who gets off the bus one morning. Eighteen students voted for May 2, 39 students voted for May 9, and 13 students voted for May 16. Which date is it likely that 105 of the 350 students would choose for the dance?
There is a probability that 105 of the 350 students would select May 9 for the dance.
What is the sample about?To solve the above we need to find the proportion of students in the sample who voted for each date and this can be done by:
May 2: 18/70
= 0.257
May 9: 39/70
= 0.557
May 16: 13/70
= 0.186
if the whole population of 350 students voted, then
May 2: 0.257 x 350
= 90
May 9: 0.557 x 350
= 195
May 16: 0.186 x 350
= 65
From the above calculation, we can see that if 105 students out of 350) were to choose a date, the most likely date that they will select is May 9, since it is the one that has the highest proportion of votes in the sample.
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Please help me with this page I’m so confused
Answer:
f(0) = 1
g(-2) = 3
f(-7)= und
g(4) x f(3) = -2 x 0 = 0
g(-4) = 2
g(x) = 0 --> x = 6, 0.5
f(x) = -1 --> x = -3, 5
f(g(3)) = f(-3) = -1
g(f(-2) = g(0) = -3
f(g(1)) = f(-3) = -1
f(g(5)) = f(-1) = 1
g(f(-4)) = g(-2) = 2
g(g(-6)) = g(4) = -2
g(f(0)) = g(1) = -3
g(f(-6)) = und
Step-by-step explanation:
In order to find the first group, such as f(0), you want to look at the f graph and find 0 on the x-axis. Wherever the y coordinate is will be the correct answer.
To find one such as f(g(3)), you want to dissect it like it is 2 problems. First, we want to find g(3) which is -3. Then we will find -3 on the f graph and find the answer with that y-coordinate.
What are the Actual dimensions of the house(in ft)
The house's real measurements are 18 feet by 20 feet.
What do we mean by dimensions?In everyday speech, a dimension is a measurement of an object's length, width, and height, such as a box.
The idea of dimension in mathematics is an expansion of the concepts of one-dimensional lines, two-dimensional planes, and three-dimensional space.
Examples of dimensions include width, depth, and height.
One dimension is that of a line, two dimensions are those of a square, and three dimensions are those of a cube. (3D).
So, scaling is the process of changing a figure's size to produce a picture.
Considering that a scale of 6 cm equals 12 ft.
Hence:
9 cm = 9 cm * (12 ft. per 6 cm) = 18 feet
10 cm = 10 cm * (12 ft. per 6 cm) = 20 feet
Therefore, the house's real measurements are 18 feet by 20 feet.
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Correct question:
A scale drawing of a house shows 9cm x10cm. If 6cm=12 ft, what are the actual dimensions?
True or false.
Solids with curves (such as cylinders) cannot be laid out as a net.
Answer:
Step-by-step explanation:
False.
Solids with curves can be laid out as a net. In fact, there are many solids with curves that can be represented by a net. For example, a cylinder can be represented by a net consisting of two circles for the top and bottom faces and a rectangle wrapping around the circumference of the circles for the side face. Similarly, a cone can be represented by a net consisting of a circle for the base and a sector of a circle for the lateral face, which can be unfolded and attached to the circular base.
While it may be more difficult to visualize and create a net for a solid with curves compared to a solid with flat faces, it is still possible to do so in many cases.
The box plot displays the number of push-ups completed by 9 students in a PE class.
A box plot uses a number line from 12 to 58 with tick marks every 2 units. The box extends from 27.5 to 42.5 on the number line. A line in the box is at 37. The lines outside the box end at 15 and 55. The graph is titled Push-Ups In PE and the line is labeled Number of Push-Ups.
Which of the following represents the value of the lower quartile of the data?
27.5
37
42.5
55
Given that 27.5 is the bottom end of the box, the lower quartile's value equation (Q1) must fall between 27.5 and 30. Hence, the appropriate response is 37.
Since, A mathematical equation links two statements and utilizes the equals sign (=) to indicate equality.
In algebra, an equation is a mathematical assertion that proves the equality of two mathematical expressions.
For instance, in the equation 3x + 5 = 14, the equal sign separates the numbers by a gap. A mathematical formula may be used to determine how the two sentences on either side of a letter relate to one another. The logo and the particular piece of software are usually identical. like, for instance, 2x - 4 = 2.
Here, 25% of the data fall inside the lower quartile (Q1), which is represented by that number. Q1 is situated near the bottom of the box in the box plot.
According to the description, the box's boundary is at 30, and its size ranges from 27.5 to 42.5 on the number line. As a result, the median value of the middle 50% of the data is 37, with a range of 27.5 to 42.5.
There are some data points outside the middle 50% since the lines outside the box finish at 15 and 55.
Hence; 27.5 is the bottom end of the box, the lower quartile's value (Q1) must fall between 27.5 and 42.5.
Hence, the appropriate response is,
= 37
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A municipality has budgeted r80 000 for putting up new street name boards the street name boards cost r134 each. how many new street name boards can be put up, and how much money will be left in the budget?
we first need to find out how many street name boards can be purchased with a budget of R80,000. We can do this by dividing the budget by the cost of each street name board:
R80,000 ÷ R134 = 597.01
Since we cannot purchase a fraction of a street name board, we round down to the nearest whole number. Therefore, the municipality can purchase 597 new street name boards with a budget of R80,000.
To find out how much money will be left in the budget, we can subtract the cost of the street name boards from the initial budget:
R80,000 - (597 x R134) = R1,598
Therefore, the municipality will have R1,598 left in their budget after purchasing 597 new street name boards.
Street name boards are an important part of any municipality as they help residents navigate and locate specific areas. By budgeting for and installing new street name boards, the municipality is taking a proactive approach to improve their community's infrastructure and ensure the safety and well-being of their residents.
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The talk-time battery life of a group of cell
phones is normally disributed with a mean of 5
hours and a standard deviation of 15 minutes.
a)what percent of the phones have a battery life of at least 4 hours and 45 minutes? b)what percent of the phones have a battery life between 4. 5 hours and 5. 25 hours? c)what percent of the phones have a battery life less than 5 hours of greater than 5. 5 hours?
a) Approximately 79.38% of the phones have a battery life of at least 4 hours and 45 minutes.
b) Approximately 34.13% of the phones have a battery life between 4.5 hours and 5.25 hours.
c) Approximately 50% of the phones have a battery life less than 5 hours or greater than 5.5 hours.
a) What percentage battery life of 4 hours and 45 minutes?
a) For phones with a mean battery life of 5 hours and a standard deviation of 15 minutes, we can calculate the percentage of phones with a battery life of at least 4 hours and 45 minutes. By converting 4 hours and 45 minutes to minutes (4*60 + 45 = 285 minutes) and using the z-score formula, we find that the z-score is (285 - 300) / 15 = -1. Hence, the percentage is approximately 1 - 0.8359 = 0.1641, which is about 16.41%.
b) What percentage battery life between 4.5 hours and 5.25 hours?
b) To determine the percentage of phones with a battery life between 4.5 hours and 5.25 hours, we need to calculate the z-scores for both values. Converting the hours to minutes, we have 4.5 hours = 270 minutes and 5.25 hours = 315 minutes. The z-scores are (270 - 300) / 15 = -2 and (315 - 300) / 15 = 1. By referring to the standard normal distribution table, we find that the area between -2 and 1 is approximately 0.6141. Thus, the percentage is 0.6141 * 100 = 61.41%.
c) What percentage battery life less than 5 hours?c) For the percentage of phones with a battery life less than 5 hours or greater than 5.5 hours, we need to calculate the z-score for both cases. The z-score for 5 hours is (300 - 300) / 15 = 0, and the z-score for 5.5 hours is (330 - 300) / 15 = 2. By referring to the standard normal distribution table, we find that the area to the left of 0 is 0.5 and the area to the right of 2 is 1 - 0.9772 = 0.0228. Adding these percentages, we get 0.5 + 0.0228 = 0.5228, which is approximately 52.28%.
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Charles draws △PQR, with m∠QPR = 130°, m∠PQR = 30°, and m∠PRQ = 20°.
Which triangle represents Charles’s triangle??
Answer:
Please look at the picture provided to see the correct triangle because each figure has no a,b, or c. Thanks
find the next three terms in the sequence 3/4, 1/2, 1/4, 0
Answer:
[tex]\sf \bf \dfrac{-1}{4} \ ; \ \dfrac{-1}{2} \ ; \ \dfrac{-3}{4}[/tex]
Step-by-step explanation:
Arithmetic sequence:
Each term in the arithmetic sequence is obtained by adding or subtracting a common number with the previous term.
To find the next three terms, we need to find the common difference.
Common difference = second term - first term
[tex]\sf = \dfrac{1}{2}-\dfrac{3}{4}\\\\=\dfrac{2-3}{4}\\\\=\dfrac{-1}{4}\\\\\\\text{Each term is obtained by adding $\dfrac{-1}{4} $ with the previous term}[/tex]
Next three terms are,
[tex]\sf 0 + \left(\dfrac{-1}{4}\right)= 0 - \dfrac{1}{4}=\dfrac{-1}{4}\\\\\\\dfrac{-1}{4}+\left(\dfrac{-1}{4}\right)=\dfrac{-1}{4}-\dfrac{1}{4}=\dfrac{-2}{4}=\dfrac{-1}{2}\\\\\\\dfrac{-1}{2}+\left(\dfrac{-1}{4}\right)=\dfrac{-1}{2}-\dfrac{-1}{4}=\dfrac{-2-1}{4}=\dfrac{-3}{4}[/tex]
A statistics class weighed 20 bags of grapes purchased from the store. The bags are advertised to contain 16 ounces, on average. The class calculated the 90% confidence interval for the true mean weight of bags of grapes from this store to be (15. 875, 16. 595) ounces. What is the sample mean weight of grapes, and what is the margin of error?
The sample mean weight is 15. 875 ounces, and the margin of error is 16. 595 ounces.
The sample mean weight is 16. 235 ounces, and the margin of error is 0. 360 ounces.
The sample mean weight is 16. 235 ounces, and the margin of error is 0. 720 ounces.
The sample mean weight is 16 ounces, and the margin of error is 0. 720 ounces
0.180 is the sample mean weight of grapes, and what is the margin of error
The sample mean weight is the midpoint of the confidence interval:
sample mean = (lower limit + upper limit) / 2 = (15.875 + 16.595) / 2 = 16.235
Therefore, the sample mean weight of grapes is 16.235 ounces.
The margin of error is half of the width of the confidence interval:
margin of error = (upper limit - sample mean) = (16.595 - 16.235) / 2 = 0.180
Therefore, the margin of error is 0.180 ounces.
So the correct answer is: "The sample mean weight is 16.235 ounces, and the margin of error is 0.180 ounces."
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The volume of a cone is 2700π cm^3. The diameter of the circular base is 30. What is the height of the cone.
Answer: 36
Step-by-step explanation:
[tex]\frac{1}{3} \pi 15^{2} h=2700\pi \\75h=2700\\h=36[/tex]
solve the equation |4x +9|=|6-5x|
|4x +9|=|6-5x|
4x+5x=6-9
9x=-3
x=-3/9
x=-1/3
Answer:
x = 15
x = - ⅓
Step-by-step explanation:
|4x+9| = |6-5x|
4x+9=6-5x or 4x+9=-6+5x
Lets do the first one first
4x+9=6-5x
9x=-3
x= -3/9 = -1/3
4x+9=-6+5x
15=x
So the two solutions are x = 15 and x = -⅓
18. A singer sells a single as a music download and CD, making a total profit of £246. 64.
She sells 456 CD singles, earning 35p for every single sold.
She earns 17p for each music download of the single.
How many music downloads did she sell?
The singer sold approximately 512 music downloads for 17p and 456 CDs for 35p and earned a total profit of £246. 64.
Given data:
Total profit = £246. 64
The price for each music download is = 17p
Number of CDs sold = 456 CD
Price for each CD = 35p
Assume that the singer sold x music downloads. when she earns 17p for each music download then her total earnings from music downloads will be 0.17x. The total earnings from selling the CD are,
= 0.35 × 456
= 159.6.
From the above data, we can write the equation to find the value of x by,
0.17x + 159.6 = 246.64
0.17x = 246.64 - 159.6
0.17x = 87.04
x = 87.04/0.17
x = 512
Therefore, the singer sold approximately 512 music downloads.
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help i need this done pls 50 points
The length of the diagonal is 12. 7in
How to determine the lengthTo determine the length of the diagonal, we need to know the Pythagorean theorem.
The Pythagorean theorem states that the square of the hypotenuse side is equal to the sum of the squares of the other two sides of a triangle.
The other two sides are the opposite and the adjacent sides.
From the information given in the diagram, we have that;
The opposite side = 12in
The adjacent side = 4in
Substitute the values
x² = 12² + 4²
find the squares
x² = 160
find the square root
x = 12. 7 in
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Your local high school is putting on a musical. They have sold 1000 tickets. Adult tickets
were sold for $8. 50, while child tickets were sold for $4. 50. A total of $7100 was
collected from ticket sells. How many tickets of each kind were sold?
650 adult tickets and 350 child tickets were sold for the local high school musical.
To determine how many adult and child tickets were sold for the local high school musical, you can use a system of linear equations. Let's use the terms "x" for adult tickets and "y" for child tickets.
1. The total number of tickets sold is 1000, so we have:
x + y = 1000
2. The total amount collected from ticket sales is $7100, so we have:
8.50x + 4.50y = 7100
Now, let's solve this system of equations step-by-step:
Step 1: Solve the first equation for x:
x = 1000 - y
Step 2: Substitute the expression for x from Step 1 into the second equation:
8.50(1000 - y) + 4.50y = 7100
Step 3: Simplify the equation:
8500 - 8.50y + 4.50y = 7100
Step 4: Combine like terms:
-4.00y = -1400
Step 5: Divide both sides by -4.00:
y = 350
Step 6: Substitute the value of y back into the expression for x from Step 1:
x = 1000 - 350
Step 7: Calculate the value of x:
x = 650
So, 650 adult tickets and 350 child tickets were sold for the local high school musical.
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Suppose f'(x) = 8x³ + 12x + 2 and f(1) = -4. Then f(-1) equals (Enter a number for your answer.)
If f'(x) = 8x³ + 12x + 2 and f(1) = -4, f(-1) is equal to -18.
Given that f'(x) = 8x³ + 12x + 2, we can find the original function f(x) by integrating f'(x) with respect to x:
f(x) = 2x⁴ + 6x² + 2x + C, where C is an arbitrary constant.
We can then use the given initial condition f(1) = -4 to solve for C:
f(1) = 2(1)⁴ + 6(1)² + 2(1) + C = -4
Simplifying, we get:
C = -16
Therefore, the function f(x) is:
f(x) = 2x⁴ + 6x² + 2x - 16
To find f(-1), we substitute x = -1 into the expression for f(x):
f(-1) = 2(-1)⁴ + 6(-1)² + 2(-1) - 16 = -18
Thus, f(-1) equals -18.
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Can someone help me with this question and show the steps please
In 0, MCA = 100° and AB CD. Also, the center of the circle, point O, is the intersection of CB and AD. What is m<1
The required value of te angle ∠3 = 100 degree.
What is Circle?A circle is a shape that is made up of all of the points in a plane that are at a certain distance from the center. Alternatively, it is the plane-moving curve traced by a point at a constant distance from a given point.
According to question:From the figure of the circle, we can identify that AD and CD are two diameter of the circle.
and ∠COA = ∠3 is inscribed angle made by up by 2 radian of the circle
So, arcCA = ∠3 = 100 degree
Thus, required value of te angle ∠3 = 100 degree.
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(20. 60 S-AQ) To study the metabolism of insects, researchers fed cockroaches measured amounts of a sugar solution. After 2, 5, and 10 hours, they dissected some of the cockroaches and measured the amount of sugar in various tissues. Five roaches fed the sugar D-glucose and dissected after 10 hours had the following amounts (in micrograms) of D-glucose in their hindguts: Amounts of D-glucose 55. 95 68. 24 52. 73 21. 5 23. 78
The mean amount of D-glucose in cockroach hindguts is 44.24 micrograms, with a standard deviation of 26.16 micrograms. The 96% confidence interval for the mean is (20.19, 68.29) micrograms.
We will first calculate the mean, standard deviation, and then the 96% confidence interval for the given data on D-glucose amounts in cockroach hindguts.
Data: 55.95, 68.24, 52.73, 21.5, 23.78
1. The mean is calculated as follows.
Mean = (55.95 + 68.24 + 52.73 + 21.5 + 23.78) / 5 = 221.2 / 5 = 44.24
The mean amount of D-glucose in cockroach hindguts is approximately 44.24 micrograms.
2. To find the standard deviation, we calculate the variance first:
Variance = [(55.95 - 44.24)^2 + (68.24 - 44.24)^2 + (52.73 - 44.24)^2 + (21.5 - 44.24)^2 + (23.78 - 44.24)^2] / (5 - 1) = 2739.736 / 4 = 684.934
Standard Deviation = √684.934 = 26.16 (rounded to two decimal places)
The standard deviation of D-glucose in cockroach hindguts is approximately 26.16 micrograms.
3. To calculate the 96% confidence interval, we need to find the margin of error:
Margin of Error = (Critical value * Standard Deviation) / √sample size
For a 96% confidence interval, the critical value (z-score) is approximately 2.05.
Margin of Error = (2.05 * 26.16) / √5 ≈ 24.05
Now, we can find the confidence interval:
Lower limit: Mean - Margin of Error = 44.24 - 24.05 = 20.19
Upper limit: Mean + Margin of Error = 44.24 + 24.05 = 68.29
The 96% confidence interval for the mean amount of D-glucose in cockroach hindguts approximately is (20.19, 68.29) micrograms.
Note: The question is incomplete. The complete question probably is: To study the metabolism of insects, researchers fed cockroaches measured amounts of a sugar solution. After 2, 5, and 10 hours, they dissected some of the cockroaches and measured the amount of sugar in various tissues. Five roaches fed the sugar D-glucose and dissected after 10 hours had the following amounts (in micrograms) of D-glucose in their hindguts: Amounts of D-glucose 55. 95, 68. 24, 52. 73, 21. 5, 23. 78. The researchers gave a 96% confidence interval for the mean amount of D-glucose in cockroach hindguts under these conditions. The insects are a random sample from a uniform population grown in the laboratory. We therefore expect responses to be Normal. What is the mean, standard deviation, and confidence interval?
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Misty needs 216 square inches of metal
to make a yield sign. If the height of the sign is 18 inches,
how long is the top edge of the sign?
24 inches
12 inches
198 inches
22 inches
pls give explanation not just the awnser
The answer is 96 inches, which is equivalent to 8 feet.
Find out the length of the top edge of the yeild sign ?To find the length of the top edge of the yield sign, we need to use the formula for the area of a trapezoid:
A = (b1 + b2)h/2
where A is the area of the trapezoid, h is the height, b1, and b2 are the lengths of the two parallel bases of the trapezoid.
In this case, we are given the area of the sign (216 square inches) and the height (18 inches), but we don't know the length of either base. However, we do know that the shape of a yield sign is that of a regular octagon, which means it has eight equal sides and eight equal angles.
If we draw a line from the top of the sign to the midpoint of one of the sides, we will form a right triangle with the height of the sign as one leg, half the length of the top edge as the other leg, and the length of one of the sides as the hypotenuse. We can use the Pythagorean theorem to find the length of the side:
a^2 + b^2 = c^2
where a is the height of the sign (18 inches), b is half the length of the top edge (what we are trying to find), and c is the length of one of the sides.
Since the sign has eight sides, we can divide the total area by 8 to get the area of one of the eight triangles that make up the sign. We can then use this area to find the length of one of the sides:
A = (bh)/2
216 sq. in. = (bh)/2
432 sq. in. = bh
Since the sign is a regular octagon, each of the eight triangles has the same base (the side of the octagon) and height (half the length of the top edge), so we can use this equation to solve for b:
432 sq. in. = b(18 in.)/2
b = 48 in.
Now we know that half the length of the top edge is 48 inches, so the full length of the top edge is:
2(48 in.) = 96 in.
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A mouse is moving through a maze and must make four turns where it can go either left or
right. The mouse will escape the maze if it makes three lefts and one right, in any order.
(a) To the right, draw a tree diagram
of all possible routes the mouse
could take.
(b) Using your tree diagram, create
an organized list of the routes. For
example, a route of right, left, left,
right could be listed as RLLR.
(C) What is the probability the mouse
escapes the maze if all turns are
randomly made?
Hence, 25% is the likelihood that the mouse will succeed in escaping the maze if all turns are made at random.
what is probability ?The examination of random chance and the likelihood that they will occur is the focus of the mathematical field of probability. It is a gauge of how likely an event is to occur and is represented by a value between zero and 1. A probability of 1 indicates that an event will undoubtedly occur. The probability of an occurrence is zero if it cannot occur. An event's probability is 0.5, or 50%, when it possesses a 50/50 chance of occurring. The number of favourable outcomes is divided by the entire amount of possible results to determine probability.
given
Based on the tree diagram, the following is an orderly list of every route that might be taken:
Three left turns and one right turn, in whatever order, make up each route.
As there are two options (left or right) for each turn, there are a total of 24 = 16 potential sequences of four turns.
Only if the mouse makes precisely three left turns and one right out of these will it be able to escape the maze.
Three lefts and one right can be arranged in one of four distinct ways (LLL, LLR, LRL, RLL), so the likelihood that the mouse will elude the maze is:
P(escape) = Number of favourable results / Number of potential results = 4 / 16 = 0.25 = 25%
Hence, 25% is the likelihood that the mouse will succeed in escaping the maze if all turns are made at random.
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A regular pentagonal prism has an edge length 9 m, and height 13 m. Identify the volume of the prism to the nearest tenth
The volume of the regular pentagonal prism with an edge length of 9 m and a height of 13 m is approximately 1811.6 m³ to the nearest tenth.
To find the volume of a regular pentagonal prism with an edge length of 9 m and a height of 13 m, follow these steps:
Step 1: Find the apothem (a) of the base pentagon. Use the formula a = s / (2 * tan(180/n)), where s is the edge length and n is the number of sides (5 for a pentagon).
a = 9 / (2 * tan(180/5))
a ≈ 6.1803 m
Step 2: Calculate the area (A) of the base pentagon. Use the formula A = (1/2) * n * s * a.
A = (1/2) * 5 * 9 * 6.1803
A ≈ 139.3541 m²
Step 3: Determine the volume (V) of the pentagonal prism. Use the formula V = A * h, where h is the height.
V = 139.3541 * 13
V ≈ 1811.6033 m³
So, the volume of the regular pentagonal prism with an edge length of 9 m and a height of 13 m is approximately 1811.6 m³ to the nearest tenth.
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Amelie has 385 muffins that she must package into boxes. Each box must hold 9 muffins. Amelie divides 385 by 9 and gets an answer of 42 R 7. What is the correct interpretation of R 7 for this situation?
In Amelie's situation, the remainder of 7 indicates that she has 7 muffins left over that cannot be packed into a full box of 9.
When Amelie divides the total number of muffins (385) by the number of muffins per box (9), she obtains a quotient of 42 and a remainder of 7. The quotient represents the number of complete boxes that Amelie can fill with 9 muffins each, while the remainder represents the number of muffins that cannot be put into a full box.
In other words, the quotient tells us how many full boxes Amelie can pack, and the remainder tells us how many muffins are left over after packing all the full boxes. In this case, Amelie can pack 42 full boxes, each with 9 muffins, which totals to 378 muffins. The remaining 7 muffins cannot fill a full box, and they are left over after all the full boxes are packed.
The "R 7" notation is commonly used to indicate the remainder in long division problems. The letter "R" stands for "remainder," and the number following it (in this case, 7) represents the actual remainder. The remainder is important because it indicates how many items are left over after dividing them into equal-sized groups.
In Amelie's situation, the remainder of 7 indicates that she has 7 muffins left over that cannot be packed into a full box of 9.
Overall, interpreting the "R 7" in this problem helps us understand how many full boxes of muffins Amelie can pack and how many muffins are left over after packing the full boxes.
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The circumstances of the base the cone is 60π cm. If the volume of the cone is 21,600π cm cubed, what is the height?
Answer:
h = 24 cm
Step-by-step explanation:
Given:
C (base) = 60π cm
V (volume) = 21,600π cm^3
Find: h (height) - ?
[tex]c = 2\pi \times r[/tex]
[tex]2\pi \times r = 60\pi[/tex]
[tex]2r = 60[/tex]
[tex]r = 30[/tex]
We found the length of the radius
v = 1/3 × πr^2 × h
1/3 × π × 900 × h = 21600π
Multiply both sides by 3:
2700π × h = 64800π / : 2700π
h = 24 cm
The dot plot shows the number of magazines bought in a month by 21 people in a office:
Dot plot labeled Number of Magazines Bought shows 10 dots over 0, 7 dots over 1, 2 dots over 2, 1 dot over 3, and 1 dot over 9
Is the median or the mean a better center for this data and why?
Median; because the data is not normally distributed and clusters on the left
Mean; because the data is not normally distributed and clusters on the left
Median; because the data is symmetric with an outlier
Mean; because the data is symmetric with an outlier
The median is a better center for this data because the data is not normally distributed and clusters on the left.
Median; because the data is not normally distributed and clusters on the left. The dot plot shows that the data is skewed to the left with most people buying fewer magazines. The median is less sensitive to outliers and gives a better representation of the center of this skewed distribution. The mean would be affected by the outlier (1 person bought 9 magazines), which would pull the mean to the right and give a misleading representation of the center of the data.
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What is the lateral area of the cone to the nearest whole number? The figure is not drawn to scale.
*
Captionless Image
34311 m^2
18918 m^2
15394 m^2
28742 m^2
Answer:
π(70)(√(70^2 + 50^2)) = π(700√74) m^3
= 18,918 m^3
Without multiplying order the products from least to greatest
Answer:
Step-by-step explanation:
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A segment with endpoints A (2, 6) and C (5, 9) is partitioned by a point B such that AB and BC form a 3:1 ratio. Find B.
A. (2. 33, 6. 33)
B. (3. 5, 10. 5)
C. (3. 66, 7. 66)
D. (4. 25, 8. 25)
To find the coordinates of point B, we can use the section formula which states that the coordinates of the point that divides a segment with endpoints (x1, y1) and (x2, y2) in the ratio of m:n are given by:
((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n))
The coordinates of point B are (4.25, 8.25), and the answer is (D).
Here, A (2, 6) and C (5, 9) are the endpoints of the segment, and we want to partition the segment in the ratio of 3:1. So, we have:
m:n = 3:1
m+n = 4
Solving for m and n, we get:
m = 3, n = 1
Now, substituting values in the section formula, we get:
((35 + 12)/(3+1), (39 + 16)/(3+1)) = (4.25, 8.25)
Therefore, the coordinates of point B are (4.25, 8.25), and the answer is (D).
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Answer:
(4.25, 8.25)
Step-by-step explanation:
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