Answer:
D. 67 square inches
Step-by-step explanation:
The area of a 16-inch pizza is:
A = πr^2 = π(8)^2 = 64π
Since the pizza was cut into 12 equal slices, each slice has an area of:
64π / 12 = 16π/3
Each boy ate 4 slices, so the total area of pizza each boy ate is:
4(16π/3) = 64π/3
Rounding to the nearest whole square inch, we have:
64π/3 ≈ 67 square inches
Therefore, the answer is D. Each boy ate approximately 67 square inches of pizza.
Hope this helps!
A model rocket is launched with an initial upward velocity of 65. The rocket's height h (in meters) after t seconds is given by the following h=65t-5t. Find all values of for which the rocket's height is 30 meters.
Answer:
30 meters after 0.5 seconds
Step-by-step explanation:
To find the values of t for which the rocket's height is 30 meters, we can set h = 30 in the given equation and solve for t:
h = 65t - 5t
30 = 65t - 5t
30 = 60t
t = 30/60
t = 0.5
Therefore, the rocket's height is 30 meters after 0.5 seconds.
Consider the diagram.
Line m is a perpendicular bisector of line segment S T. Line m also contains points S and T.
Which line segment has the same measure as TQ?
The line segment that has the same measure as TQ is B. TR
What is a Line Segment?A line segment is a critical notion in spacious geometry, and it refers to a limited section within a long line that stretches only between two fixed points.
While represented by a linear path, the essence of this structure does not allow diversion or curvature from its endpoints as they dictate the direction and length inherent in each instance.
Therefore, identified through these spatial markers, magnitudes and spatial orientations which can provide clarification for mathematical applications ranging from simpler calculations to more complex problem-solving formulas used daily when assessing either physical distances or varying volume parameters within real-life environments are observed.
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Plot the numbers -1 1/6 and 17/6 on the number line below.
The number line where we plotted -1 1/6 and 17/6 is added as an attachment
Plotting -1 1/6 and 17/6 on a number lineFrom the question, we have the following parameters that can be used in our computation:
-1 1/6 and 17/6
To start with, we convert both numbers to the same form
i.e. decimal or fraction
When converted to fractions, we have
-7/6 and 17/6
This means that we can plot -7/6 at -7 and 17/6 at point 17 where the difference in each interval is 1/6
Using the above as a guide, we have the following:
The number line is attached
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can someone tell me how to find the area of a sector in a circle and if you can provide an example
Answer:
Step-by-step explanation:
Sector area is a portion of the circle's area.
formula is π r² × x/360°
where x is the central angle.
For example if the radius is 9 and the central angle is 120°.
π · 9² · 120/360
3.14 · 81 · 1/3 = 84.78 units²
or in terms of pi = [tex]\frac{1}{3}[/tex] (81π)
what is an example in you professional life where you were unable to use an unknown in a situation
Pat's income is 20 % more than Adam. How much percent is Adam's income less than Pat's?
Adam's income is 16.67% less than Pat's income.
Let's assume Adam's income is $100
Then Pat's income is 20% more than Adam's income, which means Pat's income is:
$100 + $20 = $120
Now, we need to find out how much percent Adam's income is less than Pat's income. We can use the following formula to calculate the percentage decrease:
Percentage decrease = (Decrease in value / Original value) x 100
Decrease in value is the difference between Pat's income and Adam's income, which is:
$120 - $100 = $20
The original value is Pat's income, which is $120
So, the percentage decrease in Adam's income compared to Pat's is:
(20 ÷ 120) × 100
= 0.16666 × 100
= 16.666%
= 16.67% (approx)
Therefore, Adam's income is 16.67% less than Pat's income.
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There is a tank with 100L of water where 4kg of salt is dissolved. You open a faucet to add a salt solution of .6kg/L at the constant speed of 10 L/min. When do you have to close the faucet if you want the concentration of the salt solution in the tank to be .25kg/L in the tank? Find the time it takes after the faucet is open to the nearest minute.
Let's start by calculating the initial concentration of salt in the tank:
4 kg of salt is dissolved in 100 L of water, so the initial concentration of salt in the tank is:
4 kg / 100 L = 0.04 kg/L
We want to increase the concentration of salt in the tank to 0.25 kg/L by adding a salt solution of 0.6 kg/L at a constant rate of 10 L/min.
Let's assume that t is the time in minutes that the faucet has been open. During this time, the volume of water that has been added to the tank is 10t liters.
The amount of salt that has been added to the tank during this time is:
0.6 kg/L x 10 L/min x t min = 6t kg
The total amount of salt in the tank after t minutes is:
4 kg + 6t kg
The total volume of water in the tank after t minutes is:
100 L + 10t L
The concentration of salt in the tank after t minutes is:
(4 kg + 6t kg) / (100 L + 10t L)
We want this concentration to be 0.25 kg/L, so we can set up the following equation:
(4 kg + 6t kg) / (100 L + 10t L) = 0.25 kg/L
Simplifying this equation, we get:
16 kg + 24t kg = 25 L + 2.5t L
21.5t = 9 L
t = 9 L / 21.5 = 0.42 hours = 25.2 minutes (rounded to the nearest minute)
Therefore, you need to close the faucet after approximately 25 minutes to achieve a concentration of 0.25 kg/L in the tank.
Which inequality represents the situation "the temperature should be at least 40 degrees”?
Answer:
Step-by-step explanation:
The inequality that represents the situation "the temperature should be at least 40 degrees" is:
Temperature ≥ 40
This means that the temperature must be greater than or equal to 40 degrees.
A football field is 360 feet long and 160 feet wide. The principal is making an evacuation
plan for the school. How many students can the principal expect to fit on the football field
in an emergency? (Remember the expected floor space a standing person occupies is
about 2.5 sq feet) SHOW YOUR WORK
The football field is 360 feet long and 160 feet wide. To calculate the area, we multiply those 2 numbers:
[tex]360 \times 160 = 57600[/tex]
Now considering that the expected floor space a person occupies is 2.5 sq feet, we divide 57,600 by 2.5:
[tex]57600\div2.5 = 23040[/tex]
So 23,040 students can fit on the football field.
Help show your work 15 points !
Answer:
0.6
Step-by-step explanation:
It says x equals 0.6 so the answer was right there it was pretty easy
Adjacent angles. Solve this.
Answer:
< BEA
Step-by-step explanation:
An adjacent angle is an angle that is next to it.
< BEA is adjacent to < BEC
Divide the following polynomials using the long division model: (4×^4-5×^3+2×^2-X+5) / (x^2+x+1).
Part I: Express this problem using the standard format for a problem of dividend/divisor->
_______
divisor |dividend
Part II: Use this checklist to proceed through this problem:
• How many times does x^2 go into the largest term in the problem?
•write the value on top of the problem and multiply that value by ×^2+×+1
•write the product below the lowest line on your work and subtract if from what reminds in the problem
•continue this process until you fab no longer divide x^2 into what reminds in the problem
•include your remainder in the final answer
The quotient is [tex]4x^2[/tex] and the remainder is [tex]3x^2 - x + 5[/tex].
What is long division method?Long division is a mathematical method used to divide one number by another. It is typically used for division of larger numbers, and is taught as a basic arithmetic skill in elementary and middle school.
In long division, the dividend (the number being divided) is written on the top, and the divisor (the number dividing the dividend) is written on the left. The quotient (the answer to the division problem) is written on the top of the line below the dividend, with the remainder (the left-over amount after division) written on the right.
The process involves a series of steps, including dividing, multiplying, and subtracting. The steps are repeated until the dividend is fully divided, or until a specified level of precision is reached.
Long division is also used to divide polynomials, by using the coefficients of the terms in the polynomial instead of the actual numbers. This process is similar to long division of numbers, but involves additional steps to ensure that the division is performed correctly.
Now We want to divide the polynomial [tex]4x^4 - 5x^3 + 2x^2 - x + 5[/tex] by the polynomial [tex]x^2 + x + 1[/tex] using long division.
[tex]x^2 + x + 1 |[/tex] ([tex]4x^4 - 5x^3 + 2x^2 - x + 5[/tex] )
First, we divide the highest degree term of the dividend, [tex]4x^4[/tex], by the highest degree term of the divisor, [tex]x^2[/tex], to get [tex]4x^2[/tex] :
[tex]x^2 + x + 1 | 4x^4 - 5x^3 + 2x^2 - x + 5\\-4x^4 - 4x^3 - 4x^2\\-----------------------------\\x^3 + 2x^2[/tex]
Next, we multiply the divisor,[tex]x^2 + x + 1[/tex], by the quotient, [tex]4x^2[/tex], to get [tex]4x^4 + 4x^3 + 4x^2[/tex], and subtract it from the dividend:
[tex]x^2 + x + 1 | 4x^4 - 5x^3 + 2x^2 - x + 5\\-4x^4 - 4x^3 - 4x^2\\----------\\x^3 + 2x^2\\- 4x^2 - 4x - 4\\-------\\3x^2 - x + 5[/tex]
We now have a new polynomial, 3[tex]x^2 - x + 5[/tex], as the remainder. Since the degree of the remainder is less than the degree of the divisor, the division process is complete.
Therefore, the quotient is [tex]4x^2[/tex] and the remainder is [tex]3x^2 - x + 5[/tex]. We can write the original polynomial as:
[tex]4x^4 - 5x^3 + 2x^2 - x + 5 = (x^2 + x + 1)(4x^2) + (3x^2 - x + 5)[/tex]
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Given a random sample: X= 75, Sx = 24, and n = 36. Construct a 95% confidence interval and
estimate the population mean, m.
What would be the answer to this?
we estimate the population mean m to be 75.
What is confidence interval?
A confidence interval (CI) for an unknown parameter in frequentist statistics is a range of estimations. The most popular confidence level is 95%, but other levels, such 90% or 99%, are occasionally used for computing confidence intervals. The fraction of related CIs over the long run that actually contain the parameter's true value is what is meant by the confidence level. The degree of confidence, sample size, and sample variability are all factors that might affect the width of the CI. A larger sample would result in a narrower confidence interval if all other factors remained constant. A wider confidence interval would also be required by a higher confidence level and would be produced by a sample with more variability.
Since we want a 95% confidence interval, α = 0.05/2 = 0.025 and we need to find the critical value from the t-distribution with (36-1) = 35 degrees of freedom. Using a t-table or calculator, we find that t0.025,35 = 2.032.
Now, plugging in the values we have:
CI = 75 ± (2.032 * (24/√36))
CI = 75 ± (2.032 * 4)
CI = 75 ± 8.128
So the 95% confidence interval for the population mean m is (66.872, 83.128). This means we are 95% confident that the true population mean falls within this interval.
As for the estimated population mean, we can simply take the sample mean, which is X = 75.
Therefore, we estimate the population mean m to be 75.
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Determine the value, k, so that y=kcos(3x)+4sin(x) is a solution to the differential equation y’’+y=-9cos(2x)
The value of k such that y = k cos(3x) + 4sin(x) is a solution to the differential equation y’’ + y = -9 cos(2x) is 9/8.
Given a differential equation,
y’’ + y = -9 cos(2x)
The solution of the equation is,
y = k cos(3x) + 4sin(x)
Now,
y' = -3k sin (3x) + 4 cos(x)
y'' = -9k cos (3x) - 4 sin (x)
Substituting these to the given equation,
-9k cos (3x) - 4 sin (x) + k cos(3x) + 4sin(x) = -9 cos(2x)
-8k cos (3x) = -9 cos (3x).
Comparing,
-8k = -9
k = 9/8.
Hence the value of k is 9/8.
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27= a (a/0.8)
0.8
I dont really get this any help?
The two possible solutions of the given equation are a = √21.6 and a = -√21.6.
What is a quadratic equation?The maximum exponent of the variable in a quadratic equation, which is a polynomial equation of the second degree, is 2. The equation has two unique real solutions if the discriminant is positive. There is only one actual solution to the equation if the discriminant is zero. The equation has no genuine solutions if the discriminant is negative, but it can have two complex ones.
The given equation is 27 = a (a/0.8).
Multiply both sides of the equation by 0.8 thus we have:
21.6 = a²
Now, taking the square root on both sides we have:
a = √21.6 and a = -√21.6.
Hence, the two possible solutions of the given equation are a = √21.6 and a = -√21.6.
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The two possible solutions of the given equation are a = √21.6 and a = -√21.6.
What is a quadratic equation?The maximum exponent of the variable in a quadratic equation, which is a polynomial equation of the second degree, is 2. The equation has two unique real solutions if the discriminant is positive. There is only one actual solution to the equation if the discriminant is zero. The equation has no genuine solutions if the discriminant is negative, but it can have two complex ones.
The given equation is 27 = a (a/0.8).
Multiply both sides of the equation by 0.8 thus we have:
21.6 = a²
Now, taking the square root on both sides we have:
a = √21.6 and a = -√21.6.
Hence, the two possible solutions of the given equation are a = √21.6 and a = -√21.6.
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The average daily temperature, t, in degrees Fahrenheit for a city as a function of the month of the year, m, can be modeled by the equation graphed below, where m = 0 represents January 1, m = 1 represents February 1, m = 2 represents March 1, and so on. If the equation is t = a cosine (StartFraction pi Over 6 EndFraction (m + 1)) + k, what are the values of a and k?
On a coordinate plane, a curve starts at (0, 42). It increases to (5, 80) and then decreases to (11, 40).
Kaitlin is jogging from her house to school. She has gone 1/4 miles so far. Her school is 3 7/8 miles from her house. How many miles does Kaitlin still have to jog? Write your answer as a mixed number in simplest form.
Answer:
3 5/8 miles
Step-by-step explanation:
You want miles to go for a 3 7/8 mile trip after 1/4 mile has been taveled.
DifferenceThe remaining mileage is the difference between the total distance and the distance already covered.
3 7/8 -1/4 = 3 7/8 -2/8 = 3 5/8
Kaitlin still has 3 5/8 miles to jog.
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The Montreal Biosphere is a geodesic dome that surrounds an environmental
museum in Montreal, Canada. The dome has a volume of 6,132,812.5 cubic feet.
The structure is 75% of a full sphere. What is the length of its diameter?
Answer: 250 feet (approx.)
Step-by-step explanation:
The volume of the dome is given as 6,132,812.5 cubic feet, and we know that the dome is 75% of a full sphere. We can use this information to calculate the volume of a full sphere and then find the diameter of the sphere using the formula for the volume of a sphere.
The formula for the volume of a sphere is V = (4/3)πr^3, where V is the volume and r is the radius. Since the dome is 75% of a full sphere, the volume of the full sphere is (4/3)πr^3 / 0.75 = (16/3)πr^3 / 3.
Setting this equal to 6,132,812.5 and solving for r gives us r ≈ 35.1 feet.
Finally, the diameter of the sphere is 2r ≈ 70.2 feet.
Therefore, the length of the diameter of the Montreal Biosphere is approximately 250 feet (70.2 feet * (100/75)).
Cameron works at Fish Friends Aquatics. As part of his job, he feeds the fish, decorates the fish tanks, and helps customers choose which fish to buy. Here are the types of fish he has sold so far today:
betta, goldfish, neon tetra, betta, guppy, guppy, swordtail, betta, goldfish, goldfish
Based on the data, what is the probability that the next fish Cameron sells will be a goldfish?
On observing the types of fish that Cameron sold in a day, we can say that the probability that the next fish sold will be a "gold-fish" is 0.3 or 30%.
To find the probability of the next-fish Cameron sells being a goldfish, we use the formula:
⇒ probability = (number of desired outcomes)/(total number of possible outcomes),
In this case, the "desired-outcome" is selling a "goldfish", and
The total number of possible outcomes is the total number of fish sold so far.
To find the number of "gold-fish" sold, we need to count the number of times "goldfish" appears in the list : betta, goldfish, neon tetra, betta, guppy, guppy, swordtail, betta, goldfish, goldfish
We see that "goldfish" appears three times, so the number of desired outcomes is = 3.
The total number of possible outcomes is the total number of fish sold, which is = 10.
So, probability of the next fish Cameron sells being a goldfish is = 3/10 = 0.3,
Therefore, there is a 30% chance that the next fish Cameron sells will be a goldfish.
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what is the formula in finding the area of rectangel square triangle circle
gr 6
Answer
Rectangle: W x H = A Square: H x W Circle: [tex]\pi r ^{2}[/tex]
Step-by-step explanation:
Sorry if this is not what u are looking for I assumed u were talking about 2d shapes so yes here u go bye have a great day!!! :D
Answer
Rectangle: W x H = A Square: H x W Circle: [tex]\pi r ^{2}[/tex]
Step-by-step explanation:
Sorry if this is not what u are looking for I assumed u were talking about 2d shapes so yes here u go bye have a great day!!! :D
Find domain and range
Choose the equation that has solutions (5, 7) and (8, 13).
The equation with these solutions can be:
y = 2x - 3
How to find the equation?Because two solutions are given, we can assume that we have a linear equation.
A general linear equation can be written as:
y = a*x +b
Where a is the slope and b is the y-intercept.
If the linear equation passes through two known points, then the slope is equal to the quotient between the difference of the y-values and the difference of the x-values, here we will get.
a = (13 - 7)/(8 - 5)
a = 6/3
a = 2
Then the line is:
y = 2x + b
Replacing the values of the first point we will get:
7 = 2*5 + b
7 = 10 + b
7 - 10 = b
-3 = b
The equation is y = 2x - 3
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Melissa deposited $5000 into an account with a 4.8% annual interest rate, compounded monthly. Assuming that no withdrawals are made, how long will it take for the investment to grow to $5960?
Do not round any intermediate computations, and round your answer to the nearest hundredth.
Answer:
Step-by-step explanation:
We can use the formula for compound interest to solve this problem:
A = P(1 + r/n)^(nt)
where:
A = final amount
P = principal amount (initial investment)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = number of years
Plugging in the given values, we have:
5960 = 5000(1 + 0.048/12)^(12t)
Dividing both sides by 5000, we get:
1.192 = (1 + 0.048/12)^(12t)
Taking the natural logarithm of both sides, we get:
ln(1.192) = ln[(1 + 0.048/12)^(12t)]
Using the property of logarithms that ln(a^b) = b ln(a), we can simplify the right side:
ln(1.192) = 12t ln(1 + 0.048/12)
Dividing both sides by 12 ln(1 + 0.048/12), we get:
t = ln(1.192) / [12 ln(1 + 0.048/12)]
t ≈ 2.55
Therefore, it will take about 2.55 years (or 2 years and 7 months) for the investment to grow to $5960.
Hope that helps :)
A study was commissioned to find the mean weight of the residents in certain town. The study examined a random sample of 116 residents and found the mean weight to be 177 pounds with a standard deviation of 30 pounds. Determine a 95% confidence interval for the mean, rounding all values to the nearest tenth.
Using a t-distribution with 115 degrees of freedom (df = n-1), and a 95% confidence level, we can find the critical value using a t-table or calculator, which is approximately 1.98.
Then, we can calculate the margin of error (ME) using the formula:
ME = critical value x standard error
where the standard error (SE) is given by:
SE = standard deviation / sqrt(sample size)
Substituting the given values, we get:
SE = 30 / sqrt(116) ≈ 2.78
ME = 1.98 x 2.78 ≈ 5.5
Finally, we can construct the 95% confidence interval (CI) for the mean weight using the formula:
CI = sample mean ± margin of error
Substituting the given values, we get:
CI = 177 ± 5.5
CI ≈ [171.5, 182.5]
Therefore, the 95% confidence interval for the mean weight of the residents in the town is approximately [171.5, 182.5] pounds.
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[tex]\begin{align}\colorbox{black}{\textcolor{white}{\underline{\underline{\sf{Please\: mark\: as\: brillinest !}}}}}\end{align}[/tex]
[tex]\textcolor{blue}{\small\texttt{If you have any further questions,}}[/tex] [tex]\textcolor{blue}{\small{\texttt{feel free to ask!}}}[/tex]
♥️ [tex]{\underline{\underline{\texttt{\large{\color{hotpink}{Sumit\:\:Roy\:\:(:\:\:}}}}}}\\[/tex]
Sarah and her friends had a cookie stand at a
local ballgame. After the game, there was $42.00
left in the cashbox once they paid all their
expenses. Since Sarah did most of the work, she
decided she would keep 20% of the profit for
herself. Everyone else received 05% of the
remaining profits. How much did each person
receive? How much did Sarah receive?
Each person receives $1.68 and Sarah will receive $8.40 from the profit.
What is profit?Profit is the difference between total revenue and total expenses or costs incurred in a business or financial endeavour. It is the positive financial gain or advantage that results when the revenue earned from selling goods, services, or investments exceeds the expenses, costs, and taxes associated with producing or acquiring those goods, services, or investments.
According to the given information:
Let's break down the problem step by step to find out how much each person received, including Sarah.
Step 1: Calculate Sarah's share
Sarah decided to keep 20% of the profit for herself. The remaining profit after paying all expenses is $42.00. So Sarah's share would be 20% of $42.00.
20% of $42.00 = 0.20 * $42.00 = $8.40
So Sarah received $8.40 from the profit.
Step 2: Calculate the share for everyone else
Since Sarah kept her share of $8.40, the remaining profit for everyone else to share is $42.00 - $8.40 = $33.60.
Now, everyone else (excluding Sarah) is to receive 5% of the remaining profit. This means that each person will receive 5% of $33.60.
5% of $33.60 = 0.05 * $33.60 = $1.68
So each person (excluding Sarah) received $1.68 from the profit.
In summary:
Sarah received $8.40
Each person (excluding Sarah) received $1.68
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HELP asap lota points
MILKSHAKES
In addition to cones, you have decided to sell milkshakes. You ordered three
different sizes of cups shown below. Calculate the volume of each cup shown Use
314 for pl and round to the nearest hundredth
MIN
Total
SMALL
2DN%
6.5IN
Total
MEDIUM
2.5 IN
8 IN
Total
LARGE
6PN
Workspace for the Small Milkshake
(Show your work here.)
The volume of each milkshake cup is given by
small cup = 50.24 square inches
medium cup =127.56 square inches
Large cup =226.08 square inches
Shape of the milkshake cups is cylindrical.
Formula used to calculate volume of each cup = πr²h
Where 'r' is the radius of cups.
And h is the height of the cups
Radius of small milkshake cup = 2in.
height of small cup = 4in.
Volume of the small cup = π × 2² × 4
= 50.24 square inches
Radius of medium milkshake cup = 2.5in.
height of medium cup = 6.5in.
Volume of the medium cup = π × 2.5² × 6.5
= 127.56 square inches
Diameter of large cup = 6in
Radius of large milkshake cup = 3in.
height of large cup = 8in.
Volume of the large cup = π × 3² × 8
= 226.08 square inches
Therefore, the volume of each milkshake cup using given radius and height are as follow,
volume of small cup = 50.24 square inches
volume of medium cup =127.56 square inches
volume of Large cup =226.08 square inches
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(Two-Step Linear Inequalities MC)
Find the value of p in the inequality.
²p+1023
The value of p in the inequality is given as follows:
p ≥ -21/2.
How to solve the inequality?The inequality in the context of this problem is defined as follows:
2p/3 + 10 ≥ 3.
We must isolate the variable p, hence:
2p/3 ≥ -7 (the subtraction is inverse to the addition).
2p ≥ -21. (multiplication is inverse to division);
p ≥ -21/2. (division is inverse to the multiplication).
Hence the second option is the correct option.
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The probability that Lou is on time for a given class is 75 percent. If there are 32 classes during the semester, what is the best estimate of the number of times out of 32 that Lou is on time to class? Round your answer to the nearest integer.
The best estimate of the number of times that Lou is on time to class follows a binomial distribution and is equal to 24.
What is binomial distributionThe binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure.
The number of times that Lou is on time to class follows a binomial distribution, where the number of trials (classes) is n=32 and the probability of success (Lou being on time) is p = 75% = 0.75.
The expected number of times that Lou is on time to class can be calculated as:
E(X) = n × p = 32 × 0.75 = 24
Therefore, the best estimate of the number of times out of 32 that Lou is on time to class is 24, using the binomial distribution.
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Consider the series ∑n=0∞2e−n.
a. The general formula for the sum of the first n terms is S_n=__?__. Your answer should be in terms of n.
b. The sum of a series is defined as the limit of the sequence of partial sums, which means ∑n=0∞2e−n=limn→∞=(__?__)=(__?__).
c. Select all true statements (there may be more than one correct answer):
A. The series is a telescoping series (i.e., it is like a collapsible telescope).
B. The series converges.
C. The series is a geometric series.
D. The series is a p-series.
a. The general formula for the sum of the first n terms is Sₙ = 2(1 - e⁻ⁿ)/(1 - e⁻¹)
b. The sum of a series is defined as the limit of the sequence of partial sums is converges.
c. The true statement are "The series converges. and The series is a geometric series." (option B and C).
a. To find the general formula for the sum of the first n terms, we need to add the first n terms of the series. Thus, we have:
Sₙ = 2e⁰ + 2e⁻¹ + 2e⁻² + ... + 2e⁻ⁿ
We can see that this is a geometric series with a first term a=2 and a common ratio r=e⁻¹. Therefore, we can use the formula for the sum of a geometric series to find the general formula for Sₙ:
Sₙ = a(1 - rⁿ)/(1 - r)
Substituting a=2 and r=e⁻¹, we get:
Sₙ = 2(1 - e⁻ⁿ)/(1 - e⁻¹)
the limit of the sequence of partial sums is zero, the series converges.
b. To determine whether the series converges or diverges, we can take the limit of the sequence of partial sums as n approaches infinity. Thus, we have:
∑n=0∞2e−n=limn→∞ S_n= limn→∞ 2(1 - e⁻ⁿ)/(1 - e⁻¹)
To evaluate this limit, we can use L'Hopital's rule, which states that if we have an indeterminate form of the type 0/0 or infinity/infinity, we can take the derivative of the numerator and the denominator until the form is no longer indeterminate. Applying L'Hopital's rule, we get:
lim n→∞ 2(1 - e⁻ⁿ)/(1 - e⁻¹) = limn→∞ 2e⁻ⁿ/(e⁻¹) = 0
c. We can now identify whether the series is a telescoping series, a geometric series, or a p-series.
The series is a geometric series, as we already saw in part (a).
Therefore, the correct answers are (B) and (C).
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A company sells cardboard scratching blocks for cats. The block shaped like a right triangular prisim with a rectangular hole through its center. What is the total area of the blocks scratching surface.
Answer:
1052
Step-by-step explanation:
Fancy surface area.
(160 x 2) + (96 x 2) + (128 x 2) + (192 x 2)
=1052