The distance from the place Tyler meets his neighbor to his home is 450 meters.
Tyler's speed is 1.2 meters per second.
How to solvePart A:
To find the distance from the place Tyler meets his neighbor to his home, we need to find the distance when x=0, since it's the time they meet. Plug x=0 into the equation y=450-72x:
y = 450 - 72(0)
y = 450
The distance from the place Tyler meets his neighbor to his home is 450 meters.
Part B:
To find the speed of Tyler, we need to find the rate at which the distance is decreasing. This can be found by taking the derivative of the distance function with respect to time:
dy/dx = -72
Since the derivative is constant, Tyler's speed is constant, and it is 72 meters per minute. To convert it to meters per second, we need to multiply by the conversion factor (1 minute = 60 seconds):
Speed = 72 m/min * (1 min / 60 s)
Thus, the speed = 1.2 m/s
Tyler's speed is 1.2 meters per second.
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The side lengths of the base of a triangular prism are 5 meters, 8 meters, and 10 meters. the height of the prism is 16.5 meters. what is the lateral surface area of the prism in square meters? (please show work i beg you)
a)356.1 m²
b)388.9 m²
c)363.2 m²
d)379.5 m²
The lateral surface area of the triangular prism with side lengths of the base of a triangular prism are 5 meters, 8 meters, and 10 meters the height of the prism is 16.5 meters is 379.5 m²
Lateral surface area of prism = (a + b + c )h
a = base side = 5m
b = base side = 8m
c = base side = 10m
h = height = 16.5 m
Lateral surface area of prism = (5 + 8 + 10)16.5
The lateral surface area of the prism = 379.5m²
The lateral surface area of the triangular prism is 379.5 m²
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You are setting the combination on a three-digit lock. You want to use the numbers 123 but you don't care what order they are in.
6 different permutations using the number 1 , 2 , 3 can be masde for the lock .
Given,
1 , 2 , 3 numbers to be used for a three digit lock .
There are 3 options for the first digit, 2 options for the second digit, and 1 option for the third digit.
To find the total number of permutations, we can use the formula for permutations:
Permutations of n items taken r at a time, which is n!/(n-r)!.
Here,
In this case,
n is 3
r is 3,
So the total number of permutations is 3!/(3-3)! = 3! = 3 x 2 x 1 = 6.
Hence,
So, you can make 6 different permutations using the numbers 1, 2 and 3 in any order.
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Help please! This is for a grade... (35 points)
Hugo rides the bus to work every day, a distance of 18 miles. The distance on the route map between the station by Hugos house and the station by his work is 3 inches. What is the maps scale? 1 inch = _____ miles.
The maps scale is 1 inch = 6 miles. This means that for every inch on the map, it represents 6 miles in real-life distance.
To determine the scale, we need to know the relationship between the distance on the map and the actual distance. In this case, the distance on the route map between Hugo's house and his work is 3 inches, and the actual distance he travels by bus is 18 miles.
To find the scale, we can set up a proportion using the given information:
(distance on the map in inches) / (actual distance in miles) = (1 inch) / (x miles)
Now, we can plug in the known values:
(3 inches) / (18 miles) = (1 inch) / (x miles)
To solve for x, we can cross-multiply:
3 inches * x miles = 18 miles * 1 inch
3x = 18
Now, divide both sides by 3:
x = 6
So, the scale of the map is 1 inch = 6 miles.
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select all expressions equivalent to (3^3*3^-4)^-3
The expressions that are equivalent to (3^3*3^-4)^-3 are (3^-1)^-3, 3^3 and 27
Selecting all expressions that are equivalent to (3^3*3^-4)^-3From the question, we have the following parameters that can be used in our computation:
(3^3*3^-4)^-3
Evaluating the expression in the brackets using the law of indices, we have
(3^3*3^-4)^-3 = (3^-1)^-3
Next, we open the brackets
This gives
(3^3*3^-4)^-3 = 3^(-1 *-3)
When evaluated, we have
(3^3*3^-4)^-3 = 3^3
Evaluate the exponents
This gives
(3^3*3^-4)^-3 = 27
Hence, the expressions that are equivalent to (3^3*3^-4)^-3 are (3^-1)^-3, 3^3 and 27
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A system of equations consists of at least two equations describing a problem. True or false
True because a system of equations is a set of two or more equations that describe a particular situation or problem.
How to solve system's equations?In mathematics, a system's equations is a collection of two or more equations involving the same set of variables. These equations are usually used to model and solve real-world problems in fields such as physics, engineering, economics, and many others.
For example, consider the following system of two equations:
2x + y = 5
x - y = 3
This system of equations represents a situation where we have two unknowns, x and y, and two pieces of information that relate them. To solve the system of equations, we need to find the values of x and y that satisfy both equations simultaneously.
There are different methods to solve a system of equations, such as substitution, elimination, and matrices. The choice of method depends on the complexity of the system and personal preference. Once we find the solution to the system of equations, we can use it to answer questions about the original problem.
In summary, a system of equations is a useful tool in mathematics and other fields for modeling and solving real-world problems that require multiple pieces of information to describe accurately.
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what are the first three terms of the sequence 40-2n
Answer:
1 term-38. ,2nd term-36. ,3rd tern is34
The function f(x)=3^x-3 is an exponential function containing the points (0,-2) and (2,6).
the function g(x)=-1/2f(x)+3 containing points ____
a. (0,2)
b. (0,4)
c. (-2,3)
d. (-2,2)
and ____
a. (2,0)
b. (2,6)
c. (6,2)
d. (6,6)
The function g(x)=-1/2f(x)+3 containing points (a) (0, 4) and (a) (2, 0).
The function g(x) = -1/2f(x) + 3 is obtained by applying certain transformations to the original function f(x) = 3^x - 3.
To find the points on the graph of g(x), we need to substitute the x-values from the given points into the function g(x) and determine the corresponding y-values.
Given:
Original function f(x) = 3^x - 3
Points on f(x): (0, -2) and (2, 6)
To find the points for g(x), we substitute the x-values into g(x) = -1/2f(x) + 3:
1. For the point (0, -2):
g(0) = -1/2f(0) + 3
= -1/2(-2) + 3
= 1 + 3
= 4
2. For the point (2, 6):
g(2) = -1/2f(2) + 3
= -1/2(6) + 3
= -3 + 3
= 0
Therefore, the points for the function g(x) = -1/2f(x) + 3 are:
(a) (0, 4)
and
(a) (2, 0)
Hence, the correct answer is:
(a) (0, 4) and (a) (2, 0).
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Question 11
It took Fred 12 hours to travel over pack ice from one town in the Arctic to another town 360 miles
away. During the return journey, it took him 15 hours. Assume the pack ice was drifting at a constant
rate, and that Fred's snowmobile was traveling at a constants
What was the speed of Fred's snowmobile?
The speed of Fred's snowmobile was 30 miles per hour.
This is calculated by dividing the distance traveled by the time taken for each journey, which gives a speed of 30 mph for both the outward and return journeys.
To find Fred's speed, we can use the formula speed = distance/time. We know that Fred traveled a distance of 360 miles in 12 hours on the outward journey, so his speed was 360/12 = 30 mph.
Similarly, on the return journey, he traveled the same distance of 360 miles, but it took him 15 hours, so his speed was again 360/15 = 24 mph.
However, we are asked to find his constant speed, so we take the average of the two speeds, which gives us (30 + 24)/2 = 27 mph. Therefore, Fred's snowmobile was traveling at a constant speed of 30 mph on both journeys.
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Rewrite each equation without absolute value symbols for the given values of x.
y=|2x+5|-|2x-5|
if x<-2.5 if x>2.5
if -2.5<=x<=2.5
If x > 2.5, both expressions within absolute value symbols are positive.
The equation becomes: y = (2x + 5) - (2x - 5) = 10.
How to solve
For the given intervals of x:
If x < -2.5, both expressions within absolute value symbols are negative. Thus, the equation is: y = -(2x + 5) - (-(2x - 5)) = -10.
If x > 2.5, both expressions within absolute value symbols are positive.
The equation becomes: y = (2x + 5) - (2x - 5) = 10.
If -2.5 ≤ x ≤ 2.5, the first expression is positive and the second is negative.
The equation is: y = (2x + 5) - (-(2x - 5)) = 4x.
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A major corporation is building a 4,325 acre complex of homes, offices, stores, schools, and churches in the rural community of Glen Cove. As a result of this development, the planners have estimated that Glen Clove's population (in thousands) t years from now will be given by the following function.
P(t) = (45t^2 + 125t + 200)/t^2 + 6t + 40 (a) What is the current population (in number of people) of Glen Cove?
(b) What will be the population (in number of people) in the long run?
(a) To find the current population of Glen Cove, we need to substitute t = 0 in the given function.
P(0) = (45(0)^2 + 125(0) + 200)/(0)^2 + 6(0) + 40
P(0) = 200/40
P(0) = 5
Therefore, the current population of Glen Cove is 5,000 people (since the function is in thousands).
(b) To find the population in the long run, we need to take the limit of the function as t approaches infinity.
lim P(t) as t → ∞ = lim (45t^2 + 125t + 200)/(t^2 + 6t + 40) as t → ∞
Using L'Hopital's rule, we can find the limit of the numerator and denominator separately by taking the derivative of each.
lim P(t) as t → ∞ = lim (90t + 125)/(2t + 6) as t → ∞
Now, we can just plug in infinity for t to get the population in the long run.
lim P(t) as t → ∞ = (90∞ + 125)/(2∞ + 6)
lim P(t) as t → ∞ = ∞/∞ (since the numerator and denominator both go to infinity)
We can use L'Hopital's rule again to find the limit.
lim P(t) as t → ∞ = lim 90/2 as t → ∞
lim P(t) as t → ∞ = 45
Therefore, the population in the long run will be 45,000 people (since the function is in thousands).
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Please help me with this math problem!! Will give brainliest!! :)
part a.
the percentage of eggs between 42 and 45mm is 48.48%
part b.
The median width is approximately (42+45)/2 = 43.5mm.
The median length is approximately (56+59)/2 = 57.5mm.
part c.
The width of grade A chicken eggs has a range of about 24mm.
part d.
I think its impossible to determine because we don't have the value for the standard deviation.
The second option should be correct.
What is a histogram?A histogram is described as an approximate representation of the distribution of numerical data.
part a.
From the histogram, we see that the frequency for the bin that ranges from 42 to 45mm is 4 and we have a total of 33 eggwe use this values and calculate the percentage of eggs between 42 and 45mm is 48.48%.
part b.
we have an estimation that the median of the width is 48mm and the median of the length is around 60mm.
part c.
Also from the histogram, we notice that the smallest value is around 36mm and the largest value is around 66mm, hence the width of grade A chicken eggs has a range of about 24mm.
In a histogram, the range is the width that the bars cover along the x-axis and these are approximate values because histograms display bin values rather than raw data values.
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Ofra tried to solve an equation.
3x = 4.5
3x 4.5
3
3
=
Setting up
x = 1.5 Calculating
Where did Ofra make her first mistake?
Choose 1 answer:
Setting up
B Calculating
Ofra correctly solved the equation.
If Ofra tried to solve an equation 3x = 4.5, The statement "Ofra correctly solved the equation" is correct. So, correct option is C.
We can see this by substituting x = 1.5 into the original equation 3x = 4.5:
3(1.5) = 4.5
Simplifying the left-hand side, we get:
4.5 = 4.5
This is a true statement, which means that x = 1.5 is a valid solution to the equation 3x = 4.5.
Therefore, Ofra did not make any mistakes in solving the equation. She correctly set up the equation 3x = 4.5 by multiplying both sides by 3 to isolate x, and then calculated the value of x to be 1.5, which is the correct solution.
Option (c) is the correct answer.
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Complete question is:
Ofra tried to solve an equation.
3x = 4.5, Setting up x = 1.5 Calculating
Where did Ofra make her first mistake?
Choose 1 answer:
a) Setting up
b) Calculating
c) Ofra correctly solved the equation.
Sneha’s mother is 12 years more than twice Sneha’s age. After 8 years, she will be 20 years
less than three times Sneha’s age. Find Sneha’s age and Sneha’s mother’s age.
Sneha's current age is 16 years old. Sneha's mother is 44 years old.
Let's assume Sneha's current age is x.
Sneha's mother's current age = 2x + 12
After 8 years, Sneha's age = x + 8
After 8 years, Sneha's mother's age = 2x + 12 + 8 = 2x + 20
After 8 years, Sneha's mother's age will be 20 less than three times Sneha's age: 2x + 20 = 3(x + 8) - 20
Now we can solve for x:
2x + 20 = 3(x + 8) - 20
2x + 20 = 3x + 24 - 20
2x + 20 = 3x + 4
x = 16
Therefore, Sneha's current age is 16 years old.
Sneha's mother's current age = 2x + 12
= 2(16) + 12 = 44
So, Sneha's mother is 44 years old.
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ANALYZE In Europe, fuel economy is measured in liters per 100 kilometers. On a 2500 km road trip a car used 150 liters of fuel. The car used an average of how many liters per 100 km?
To find out the car's average fuel consumption in liters per 100 kilometers, we need to use the following formula:
Fuel consumption (liters per 100 km) = (Total fuel used / Total distance traveled) x 100
Using the information given, we know that the car traveled 2500 km and used 150 liters of fuel. Plugging these values into the formula, we get:
Fuel consumption (liters per 100 km) = (150 / 2500) x 100 = 6
Therefore, the car used an average of 6 liters of fuel per 100 kilometers on its 2500 km road trip.
It is important to note that fuel economy is an important factor in Europe, where fuel prices are generally higher than in other parts of the world.
By measuring fuel consumption in liters per 100 kilometers, European consumers can easily compare the fuel efficiency of different vehicles and make more informed purchasing decisions.
Additionally, analyzing fuel consumption data can help drivers identify ways to improve their fuel efficiency, such as reducing their speed, maintaining proper tire pressure, and avoiding excessive idling.
This not only saves money on fuel costs, but also reduces carbon emissions and contributes to a more sustainable transportation system.
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Nadia bought five tickets to attend a spaghetti supper fund raiser at her school. The equation 5x = 32. 50 can be used to find X, the cost of each ticket in dollars. Which equation represents the cost of each ticket. A. X=32. 50/5
B. X=32. 50(5)
C. X= 32. 50-5
D. X= 32. 50+5
Nadia brought five tickets to attend a spaghetti supper fund rasier at her school. The equation 5x = 32.50 can be used to find x, the cost of each ticket in dollars. The equation x = 32.50/5 will represent the cost of each ticket.
This is because the equation 5x = 32.50 is asking us to find the cost of each ticket (represented by x) when there are five tickets in total and the total cost is $32.50.
To solve for x, we need to isolate it on one side of the equation. We can do this by dividing both sides by 5, which gives us:
X=32.50/5.
So, each ticket costs $6.50.
Therfore, the correct equation that represents the cost of each ticket is X=32.50/5, option A.
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You buy a movie ticket for $5.25
and popcorn for $2.98
.
You pay with a $10
bill.
can someone help me?
Answer:12
Step-by-step explanation:
Based on results from recent track meets, Leon has a 64% chance of getting a medal in the 100 meter dash. Estimate the probability that Leon will get a medal in at least 4 of the next 10 races. Use the random number table, and make at least 10 trials for your simulation. Express your answer as a percent
The estimated probability of Leon getting a medal in at least 4 of the next 10 races is 80%.
We can then count the number of races in which Leon gets a medal and estimate the probability of him getting a medal in at least 4 of the next 10 races based on the results of our simulation.
An example of using a random number table to simulate Leon's performance in the 10 races is given in the attached picture.
Based on this simulation, Leon got a medal in 5 of the 10 races. We can repeat this simulation multiple times (e.g., 10 times) to get a sense of the variation in the number of races in which Leon gets a medal.
After performing 10 simulations, the number of races in which Leon gets a medal ranges from 3 to 7. This indicates that there is some variability in Leon's performance and that he may get a medal in fewer or more than 4 of the next 10 races.
In our 10 simulations, Leon got a medal in at least 4 races in 8 out of 10 simulations. Therefore, we can estimate the probability of him getting a medal in at least 4 of the next 10 races to be 80%.
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Convert 4/5 to a decimal and a percent.
Decimal (Edit the repeating and non-repeating part):
0.00
Percent (Edit the repeating and non-repeating part)
0.00%
The decimal form of 4/5 is 0.80, and it is equivalent to 80% as a percent.
To convert the fraction 4/5 into a decimal and a percent, we start by dividing the numerator (4) by the denominator (5). The result is 0.80 as a decimal.
In decimal form, 4/5 is written as 0.80. The "0" before the decimal point represents whole units, and the "80" after the decimal point represents hundredths.
To express this decimal as a percent, we multiply it by 100, as a percent is a representation of parts per hundred. So, 0.80 multiplied by 100 equals 80. Therefore, 4/5 is equivalent to 80% when expressed as a percentage.
In summary, 4/5 as a decimal is 0.80, which means it represents 80 hundredths, and as a percent, it is 80%, which signifies 80 parts per hundred. This conversion is particularly useful in various mathematical and real-world applications, such as calculating discounts, grades, proportions, and percentages in everyday life and business contexts.
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Pls Help fast! Find the diameter of the circle
11 Points and brainliest!
Answer: 16
Step-by-step explanation: to find diameter its radius x 2, so 2 x 8 is 16
Customer: "I prepaid 70% of the total cost of $127. 50 last week. "
Employee: "Okay. The total you still owe is
The total amount the customer still owe is $38.25. Customer: "I prepaid 70% of the total cost of $127.50 last week."
Employee: "Okay. The total you still owe is $38.25."
Here the customer prepaid 70% of total cost of $127.50 last week and the employee need to replay how much amount the customer still owe is. To find out the total amount the customer still owes after prepaying 70% of the total cost of $127.50,
Calculate the prepaid amount: 70% of $127.50 = 0.7 * 127.50 = $89.25Subtract the prepaid amount from the total cost: $127.50 - $89.25 = $38.25So the total amount the customer still owe is $38.25.
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The notation (x,y)→(−x,y) means a reflection across the y axis.
Answer:
This is true.
Step-by-step explanation:
To prove this as true what we can do is draw a graph. On one of the graphs, we will have a point at (-7,1). If we were going to reflect it over the y-axis by counting the distance it is from the y-axis and counting it in the other direction. When we do this we get a point of (7,1). We can infer that because it was flipped in the y-axis the y value stayed the same while the x-axis changed.
This is how we can prove this to be true.
If f(x) = x - 7, then what is ƒ (8)?
Step-by-step explanation:
we use a function as a kind of recipe or template for an actual calculation (or actual "ingredients").
as long as we don't handle the actual food items or actual numbers, it all stays theoretical. variables have no actual value and represent every possible case with every possible value.
but as soon as we get an actual unit value (like 8 in our case), we can put it in place of the variables (that are really nothing else but placeholders for actual values) and simply caucuses the result.
so,
when the question asks what is f(8), it really means what is the result when x = 8.
therefore,
f(8) = 8 - 7 = 1
that's it. that is the whole thing. no mystery, magic or genius strikes necessary.
Answer:
[tex] \sf \: f(8) = 1[/tex]
Step-by-step explanation:
Given function,
→ f(x) = x - 7
Now we have to,
→ Find the required value of f(8).
We have to use,
→ x = 8
Then the value of f(8) will be,
→ f(x) = x - 7
→ f(8) = 8 - 7
→ [ f(8) = 1 ]
Hence, the value of f(8) is 1.
The height of the roof is 30ft and the radius of the base is 15tf. what is the area of the roof? what is the lateral surface area of the roof
The lateral surface area of the roof can be found by subtracting the area of the base from the total surface area:1581.84 sq ft
Assuming the roof is a cone:
The slant height of the cone can be found using the Pythagorean theorem:
l = √(r^2 + h^2) = √(15^2 + 30^2) = 33.541 ft
The area of the roof can be found using the formula for the surface area of a cone:
A = πr^2 + πrl = π(15)^2 + π(15)(33.541) ≈ 1800.66 sq ft
The lateral surface area of the roof can be found by subtracting the area of the base from the total surface area:
L = πrl = π(15)(33.541) ≈ 1581.84 sq ft
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The scatterplot shows the relationship between two variables, x and y, for the 9 points in data set
A. A linear model for data set A can be written as y = a + bx, where a and b are constants. Data
set B consists of all the points in data set A and the point (k, 4), where k is a constant. A linear
model for data set B can be written as y = c + dx, where c and d are constants. Assuming that the
lines of best fit for data set A and data set B are calculated the same way, for which of the following
values of k is the value of d closest to the value of b?
The slope of the line of best fit for data set B is -1.495, which is closest to the slope of the line of best fit for data set A (-1.5).
How to solve for the slopeWhen k = 4:
Σ(x) = 39, Σ(y) = 61, Σ(xy) = 566, Σ(x²) = 316
n = 10
d = (Σ(xy) - (Σx)(Σy) / n) / (Σ(x²) - (Σx)² / n) = (566 - (39)(61) / 10) / (316 - (39)² / 10) = -1.495
When k = 5:
Σ(x) = 40, Σ(y) = 65, Σ(xy) = 610, Σ(x²) = 337
n = 10
d = (Σ(xy) - (Σx)(Σy) / n) / (Σ(x²) - (Σx)² / n) = (610 - (40)(65) / 10) / (337 - (40)² / 10) = -1.481
Based on these calculations, it appears that the value of k that makes d closest to b is k = 4.
At k = 4, the slope of the line of best fit for data set B is -1.495, which is closest to the slope of the line of best fit for data set A (-1.5).
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a chain letter starts when a person sends it to 7 others. these people either ignore it or send it to 7 more. if 211 are involved in this chain letter (including the sender), (1) how many sent the letter? (2) how many did not continue the chain?
There are 22 people who sent the chain letter, and 189 people did not continue the chain.
We know that the chain started with one person who sent it to 7 others, so that makes a total of 8 people in the first round. In the second round, each of those 7 people could either send it to 7 more people or ignore it, so there are two possibilities for each of those 7 people: they either continue the chain or they don't.
Therefore, there are 2⁷ = 128 possible outcomes for the second round.
If we assume that everyone who received the letter in the second round sent it to 7 more people, then there would be 7 x 128 = 896 people in the third round.
Continuing this pattern, we can see that the number of people in each round is given by the formula of combination 8 x 7ⁿ⁻¹, where n is the round number (starting with n = 1 for the first round).
We want to find the round number such that the total number of people in the chain is 211. Setting the formula above equal to 211 and solving for n gives
8 x 7ⁿ⁻¹ = 211
7ⁿ⁻¹ = 26.375
n - 1 = log_7(26.375)
n = 2.78 (rounded to two decimal places)
Since we can't have a fractional round number, we can assume that the chain ended after the second round (since the third round would have too many people). Therefore, the total number of people who sent the letter is
8 + 7(2) = 22
To find the number of people who did not continue the chain, we can subtract the number of people who sent the letter from the total number of people in the chain
211 - 22 = 189
Therefore, 189 people did not continue the chain.
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Daria performed an experiment in which she randomly pulled a marble from a bag, recorded its color, put it back, and then repeated. The following table represents the number of times each color of marble was pulled.
Color of Marble Frequency
Yellow 26
Green 34
Purple 18
Red 22
If Daria repeats the experiment 50
more times, how many of those times should she expect to pull a yellow marble?
The number of times she should expect to pull a yellow marble is 13
How many of those times should she expect to pull a yellow marble?From the question, we have the following parameters that can be used in our computation:
Color of Marble Frequency
Yellow 26
Green 34
Purple 18
Red 22
This means that
Times she should expect to pull a yellow marble is
Yellow = P(Yellow) * 50
So, we have
Yellow = 26/(26 + 34 + 18 + 22) * 50
Evaluate
Yellow = 13
Hence, the number of times she should expect to pull a yellow marble is 13
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2 questions that I am stuck on.
8. x=(a+b)/c.
The given equation is,
(b-cx)/a+(a-cx)/b+2=0
⇒b/a-cx/a+a/b-cx/b+2=0
Taking the variables to LHS and constants to RHS,
-cx/a-cx/b=-b/a-a/b-2
or, cx/a+cx/b=b/a+a/b+2
or, cx(1/a+1/b)=b/a+a/b+2
Multiplying both sides of the above equation by ab,
or, cx(a+b)/ab=(a²+b²+2ab)/ab
⇒cx(a+b)=(a²+b²+2ab)
or, cx(a+b)=(a+b)²
∴ x=(a+b)²/c(a+b)=(a+b)/c
Hence x=(a+b)/c.
9. x= -ab(c-a+b)
The given equation is,
a/(x+a)+b/(x-b)=(a+b)/(x+c)
Multiplying the LHS and RHS of the equation by (x+a)(x-b)(x+c),
a(x-b)(x+c)+b(x+a)(x+c)=(a+b)(x+a)(x-b)
⇒a(x²-bx+cx-bc)+b(x²+ax+cx+ac)=(a+b)(x²+ax-bx-ab)
The above equation has terms with variables x²,x and constant terms.
Keeping the like terms together,
x²(a+b-a-b)+x(-ab+ac+ab+bc-a²+b²)= abc-abc-a²b-ab²
⇒ x²(0)+x(ac+bc-a²+b²)= -a²b-ab²
⇒ x = (-a²b-ab²)/(ac+bc-a²+b²)
= -ab(a+b)/[c(a+b)-(a+b)(a-b)]
= -ab(a+b)/(a+b)(c-a+b)
= -ab/(c-a+b)
Hence, x= -ab/(c-a+b)
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The solutions for questions 8 and 9 are:
8. b = (ac - 2ab)/(2-a)
9. x = [-b ± sqrt((a+b)^2 + 4b(a+c))]/(2b)
How did we get the values?To solve the equation:
(b-cx)/a+(a-cx)/b+2=0
Simplify the equation by finding a common denominator.
Multiply the first term by b/b and the second term by a/a, then add them together:
(b^2 - bcx + a^2 - acx)/(ab) + 2 = 0
collect like terms:
(b^2 + a^2)/(ab) - cx(a+b)/(ab) + 2 = 0
Multiply both sides by ab to eliminate the denominator:
b^2 + a^2 - cx(a+b) + 2ab = 0
Simplify:
cx = (a^2 + b^2 + 2ab)/(a+b)
cx = (a+b)^2/(a+b)
cx = a+b
Substitute cx with a+b:
(b-c(a+b))/a + (a-c(a+b))/b + 2 = 0
Simplify:
(2b - ac - bc)/(ab) = -2
Multiply both sides by ab:
2b - ac - bc = -2ab
Solve for b:
b = (ac - 2ab)/(2-a)
9. To solve the equation:
a/(x+a) + b/(x-b) = (a+b)/(x+c)
We can start by finding a common denominator on the left side:
(a(x-b) + b(x+a))/((x+a)(x-b)) = (a+b)/(x+c)
Simplify:
(ax - ab + bx + ab)/((x+a)(x-b)) = (a+b)/(x+c)
collect like terms:
(ax + bx)/((x+a)(x-b)) = (a+b)/(x+c)
Factor out x:
x(a+b)/((x+a)(x-b)) = (a+b)/(x+c)
Cross-multiply:
(a+b)(x+c) = x(a+b)(x-b)
Expand and simplify:
ax + bx + ac + bc = ax^2 - bx^2
Rearrange and simplify:
bx^2 + (a+b)x - (a+c)b = 0
Use the quadratic formula to solve for x:
x = [-b ± sqrt((a+b)^2 + 4b(a+c))]/(2b)
Note that this equation has a restriction on x, namely that x cannot be equal to a or b, since that would make some of the denominators zero.
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Drag each set of dots to the correct location on the dot plot. Each set of dots can be used more than once. Not all sets of dots will be used Tricia recorded the number of pets owned by each of her classmates. These data points represent the results of her survey 032. 41. 2. 1. 2. 1. 1. 3,4,2,0,0. 1. 1. 1. 0. 3 Create a dot plot that represents the data 1 Number of Students 2 Numbers of Pets
To create a dot plot that represents the given data, we need to place each data point on a number line and then represent it with a dot above its corresponding value. The number line should range from 0 to the maximum number of pets owned by any student, which in this case is 4.
First, we place a dot above the value 0, which occurs twice in the data set. Then, we place three dots above the value 1, which occurs five times in the data set. Next, we place two dots above the value 2, which occurs twice in the data set. Finally, we place one dot above the value 3 and one dot above the value 4.
In summary, the dot plot will have one dot above 3 and one dot above 4 for the number of pets axis, and for the number of students axis, we will have two dots above 0, five dots above 1, two dots above 2, and no dots above 3 or 4. This represents the distribution of the data in a visual way.
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