Answer:
40 loads
Step-by-step explanation:
To find how many loads of wash you can do you need to divide 48 by 1 1/5.
There are two ways you can divide this, the first way is converting 48 to a fraction and dividing them.
48/1 divided by 1 1/5
convert 1 1/5 to an improper fraction
48/1 divided 6/5
change the division to multiplication and find the reciprocal of the second fraction.
48/1*5/6 = 240/6
Simplify to 40/1 or 40
The second way is changing 1 1/5 to a decimal, so its 1.2
Then divide 48 by 1.2 and you get 40.
By dividing the total amount of detergent by the amount required per load, you can determine that a 48-cup family-size box of laundry detergent can do 40 loads of wash.
Explanation:To find out how many loads of wash can be done with a family-size box of laundry detergent, you need to divide the total amount of detergent, 48 cups, by the amount required per load, which is 1 and 1/5 cups.
Firstly, we need to convert the mixed fraction into an improper fraction. 1 and 1/5 = 5/5 + 1/5 = 6/5.
Then, we do the division: 48 ÷ (6/5) = 48 * (5/6) = 40. This operation is equivalent to multiplying by the reciprocal of the fraction.
So, with a 48-cup family-size box of laundry detergent, you could do 40 loads of wash, assuming each load requires 1 and 1/5 cups of detergent.
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A gambler mixed a "cheat" die with all sixes into a box of eight normal dice. She chooses one at random, rolls it twice, and gets six both times. What is the probability that she chose the "cheat" die?
Answer:
The probability is [tex]\frac{1}{9}[/tex]
Step-by-step explanation:
There are a total of 9 die in the box after she added the "cheat" die. Since there is only 1 "cheat" die in the box and chooses a die at random then the probability of her having chosen the cheat die is [tex]\frac{1}{9}[/tex] . The fact that she rolled two sixes did not affect when she choose the die therefore the probability remains as [tex]\frac{number.of.cheat.die}{total.dice}[/tex].
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For the lines x=3t, y=1-2t, z=2-3t and x (3, 1, 4) +s(-9, 6, 9) (a) Show that the lines are parallel. (b) Calculate the distance between the paralle lines.
Answer:
Step-by-step explanation:
Given lines in parametric form
line [tex]L_1[/tex]
[tex]\frac{x}{3}=\frac{y-1}{-2} =\frac{z-2}{-3}[/tex]
direction vector of [tex]L_1 v_1=<3,-2,-3 >[/tex]
Line [tex]L_2[/tex]
direction vector of [tex]L_2 v_2=<-9,6,9 >[/tex]
therefore
[tex]v_2=-3v_1[/tex]
thus lines are parallel.
(ii)distance between two lines is
[tex]L_2[/tex] is given by
[tex]\frac{x-3}{-9}=\frac{y-1}{6} =\frac{z-4}{9}[/tex]=s
[tex]\frac{x-3}{-3}=\frac{y-1}{2} =\frac{z-4}{3}[/tex]=3s
[tex]\left | \frac{\begin{vmatrix}x_2-x_1 &y_2-y_1 &z_2-z_1 \\ a_1&b_1&c_1 \\ a_2&b_2&c_2\end{vmatrix}}{\sqrt{\left ( a_1b_2-a_2b_1 \right )^2+\left ( b_1c_2-b_2c_1 \right )^2+\left ( c_1a_2-c_2a_1 \right )^2}}\right |[/tex]
where [tex]a_1[/tex]=3
[tex]b_1[/tex]=-2
[tex]c_1[/tex]=-3
[tex]a_2[/tex]=-9
[tex]b_2[/tex]=6
[tex]c_2[/tex]=9
distance(d)=0 units since value of the matrix
[tex]\begin{vmatrix}x_2-x_1 &y_2-y_1 &z_2-z_1\\ a_1&b_1&c_1 \\a_2&b_2 &c_2 \end{vmatrix}[/tex]
is zero
A particular dam contains approximately 1,100,000,000,000 cubic feet of water. For a week-long spike flood, water was released at a rate of 25,100 cubic feet per second.
The amount of water released during the week-long flood was?
Answer:
The amount of water released during the week-long flood was 15,180,400,000 cubic feet per second.
Step-by-step explanation:
How many seconds are there in a week?
Each minute has 60 seconds
Each hour has 60 minutes
Each day has 24 hours
Each week has 7 days. So
60*60*24*7 = 604,800
A week has 604,800 seconds.
Water was released at a rate of 25,100 cubic feet per second.
In a week(604,800 seconds)
604,800*25,100 = 15,180,400,000
The amount of water released during the week-long flood was 15,180,400,000 cubic feet per second.
The distribution of cholesterol levels in teenage boys is approximately normal withnbsp mu equals 170 and sigma equals 30 (Source: U.S. National Center for Health Statistics). Levels above 200 warrant attention. Find the probability that a teenage boy has a cholesterol level greater than 225.
Answer: 0.0336
Step-by-step explanation:
Given : The distribution of cholesterol levels in teenage boys is approximately normal with mean :[tex]\mu= 170[/tex]
Standard deviation : [tex]\sigma= 30[/tex]
The formula for z-score is given by :-
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x=225
[tex]z=\dfrac{225-170}{30}=1.83[/tex]
The p-value =[tex]P(z>1.83)=1-P(z<1.83)[/tex]
[tex]=1-0.966375=0.033625\approx0.0336[/tex]
The probability that a teenage boy has a cholesterol level greater than 225 =0.0336
Use the Chain Rule to find the indicated partial derivatives. u = x2 + yz, x = pr cos(θ), y = pr sin(θ), z = p + r; (partial u)/(partial p), (partial u)/(partial r), (partial u)/(partial theta) when p = 2, r = 2, θ = 0
[tex]u(x,y,z)=x^2+yz[/tex]
[tex]\begin{cases}x(p,r,\theta)=pr\cos\theta\\y(p,r,\theta)=pr\sin\theta\\z(p,r,\theta)=p+r\end{cases}[/tex]
At the point [tex](p,r,\theta)=(2,2,0)[/tex], we have
[tex]\begin{cases}x(2,2,0)=4\\y(2,2,0)=0\\z(2,2,0)=4\end{cases}[/tex]
Denote by [tex]f_x:=\dfrac{\partial f}{\partial x}[/tex] the partial derivative of a function [tex]f[/tex] with respect to the variable [tex]x[/tex]. We have
[tex]\begin{cases}u_x=2x\\u_y=z\\u_z=y\end{cases}[/tex]
The Jacobian is
[tex]\begin{bmatrix}x_p&x_r&x_\theta\\y_p&y_r&y_\theta\\z_p&z_r&z_\theta\end{bmatrix}=\begin{bmatrix}r\cos\theta&p\cos\theta&-pr\sin\theta\\r\sin\theta&p\sin\theta&pr\cos\theta\\1&1&0\end{bmatrix}[/tex]
By the chain rule,
[tex]u_p=u_xx_p+u_yy_p+u_zz_p=2xr\cos\theta+zr\sin\theta+y[/tex]
[tex]u_p(2,2,0)=2\cdot4\cdot2\cos0+4\cdot2\sin0+0\implies\boxed{u_p(2,2,0)=16}[/tex]
[tex]u_r=u_xx_r+u_yy_r+u_zz_r=2xp\cos\theta+zp\sin\theta+y[/tex]
[tex]u_r(2,2,0)=2\cdot4\cdot2\cos0+4\cdot2\sin0+0\implies\boxed{u_r(2,2,0)=16}[/tex]
[tex]u_\theta=u_xx_\theta+u_yy_\theta+u_zz_\theta=-2xpr\sin\theta+zpr\cos\theta[/tex]
[tex]u_\theta(2,2,0)=-2\cdot4\cdot2\cdot2\sin0+4\cdot2\cdot2\cos0\implies\boxed{u_\theta(2,2,0)=16}[/tex]
This problem is about using the Chain Rule to compute the partial derivatives of a function with respect to different variables, followed by substitution of specific values into the obtained derivatives.
Explanation:The problem involves finding partial derivatives using the Chain Rule on the given equations with given parameters: p = 2, r = 2, θ = 0. By substituting the equations for x, y, z into u which gives us u = (prcosθ)² + prsinθ(p+r). The next step is to compute (partial u)/(partial p), (partial u)/(partial r), (partial u)/(partial theta) by using the Chain Rule to find each partial derivative. After computing, you just substitute the given values of p, r, θ into the obtained derivates to get the final answers.
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Prove by induction that 3n(n 1) is divisible by 6 for all positive integers.
We are asked to prove by the method of mathematical induction that:
3n(n+1) is divisible by 6 for all positive integers.
for n=1 we have:[tex]3n(n+1)=3\times 1(1+1)\\\\i.e.\\\\3n(n+1)=3\times 2\\\\i.e.\\\\3n(n+1)=6[/tex]
which is divisible by 6.
Hence, the result is true for n=1
Let the result is true for n=ki.e. 3k(k+1) is divisible by 6.
Now we prove that the result is true for n=k+1Let n=k+1
then
[tex]3n(n+1)=3(k+1)\times (k+1+1)\\\\i.e.\\\\3n(n+1)=3(k+1)(k+2)\\\\i.e.\\\\3n(n+1)=(3k+3)(k+2)\\\\i.e.\\\\3n(n+1)=3k(k+2)+3(k+2)\\\\i.e.\\\\3n(n+1)=3k^2+6k+3k+6\\\\i.e.\\\\3n(n+1)=3k^2+3k+6k+6\\\\i.e.\\\\3n(n+1)=3k(k+1)+6(k+1)[/tex]
Since, the first term:
[tex]3k(k+1)[/tex] is divisible by 6.
( As the result is true for n=k)
and the second term [tex]6(k+1)[/tex] is also divisible by 6.
Hence, the sum:
[tex]3k(k+1)+6(k+1)[/tex] is divisible by 6.
Hence, the result is true for n=k+1
Hence, we may say that the result is true for all n where n belongs to positive integers.
To prove by induction that the expression 3n(n+1) is divisible by 6, we start with the base case of n=1, which is divisible by 6, and then show that if it holds for an integer k, it also holds for k+1. By factoring and using the induction hypothesis, we demonstrate the expression's divisibility by 6 for all positive integers.
Proof by Induction of Divisibility by 6
To prove by induction that 3n(n+1) is divisible by 6 for all positive integers, we follow two steps: the base case and the inductive step.
Base Case
Let's check for n=1:
Inductive Step
Assume the statement holds for a positive integer k, so 3k(k+1) is divisible by 6.
Now, we must show that 3(k+1)((k+1)+1) = 3(k+1)(k+2) is also divisible by 6.
Factoring out the common term we get:
Notice that (k+1) is an integer, hence 3(k+1) is divisible by 3. If k is even, then k+1 is odd, so 3(k+1) is still divisible by 3 but not necessarily by 6. However, if k is odd, k+1 is even and 3(k+1) is divisible by 6. Since the divisible by 3 part is always true, and the divisible by an additional factor of 2 part is true every other time, the sum 6m + 3(k+1) is divisible by 6 regardless of whether k is odd or even. This is because adding a multiple of 6 to either another multiple of 6 or a multiple of 3 always results in a multiple of 6.
Therefore, 3n(n + 1) is divisible by 6 for all positive integers n.
Let R be a communtative ring and a, b elements in R. Prove that if a and b are units, then so is ab. What can we say about ab when a is a unit and b is a zero divisor? Prove your claim.
Answer with explanation:
Let R be a communtative ring .
a and b elements in R.Let a and b are units
1.To prove that ab is also unit in R.
Proof: a and b are units.Therefore,there exist elements u[tex]\neq0[/tex] and v [tex]\neq0[/tex] such that
au=1 and bv=1 ( by definition of unit )
Where u and v are inverse element of a and b.
(ab)(uv)=(ba)(uv)=b(au)(v)=bv=1 ( because ring is commutative)
Because bv=1 and au=1
Hence, uv is an inverse element of ab.Therefore, ab is a unit .
Hence, proved.
2. Let a is a unit and b is a zero divisor .
a is a unit then there exist an element u [tex]\neq0[/tex]
such that au=1
By definition of unit
b is a zero divisor then there exist an element [tex]v\neq0[/tex]
such that bv=0 where [tex]b\neq0[/tex]
By definition of zero divisor
(ab)(uv)=b(au)v ( because ring is commutative)
(ab)(uv)=b.1.v=bv=0
Hence, ab is a zero divisor.
If a is unit and b is a zero divisor then ab is a zero divisor.
Of the 122 students who took a mathematics exam, 76 correctly answered the first question, 60 correctly answered the second question, and 38 correctly answered both questions. How many students answered the first question correctly, but not the second? students answered the first question correctly, but not the second
Answer: 38
Step-by-step explanation:
If P and Q are two different set the their difference is given by P-Q i.e. the number of elements in P bit not Q .i.e. [tex]P-Q=n(P)-n(P\cap Q)[/tex]
Let A be the number of students who correctly answered the first question and B be the number of students who correctly answered the second question .
Given : [tex]n(A)=76[/tex]
[tex]n(B)=60[/tex]
[tex]n(A\cap B)=38[/tex]
Then the number of students who answered the first question correctly, but not the second is given by :-
[tex]A-B=n(A)-n(A\cap B)\\\\=76-38=38[/tex]
Hence, the number of students who answered the first question correctly, but not the second is 38.
Find the area of the region enclosed by the graphs of x=10-y^2 and x=7 (Use symbolic notation and fractions where needed.)
Answer:[tex]18\sqrt{3}[/tex]
Step-by-step explanation:
Given data
we haven given a parabola and a straight line
Parabola is [tex]{y^2}={-\left ( x-10\right )[/tex]
line is [tex]x=7[/tex]
Find the point of intersection of parabola and line
[tex]y=\pm \sqrt{3}[/tex] when[tex]x=7[/tex]
Area enclosed is the shaded area which is given by
[tex]Area=\int_{0}^{\sqrt{3}}\left ( 10-y^2 \right )dy[/tex]
[tex]Area=_{0}^{\sqrt{3}}10y-_{0}^{\sqrt{3}}\frac{y^3}{3}[/tex]
[tex]Area=10\sqrt{3}-\sqrt{3}[/tex]
[tex]Area=9\sqrt{3}units[/tex]
Required area will be double of calculated because it is symmetrical about x axis=[tex]18\sqrt{3}units[/tex]
To find the area of the region enclosed by the graphs of[tex]x=10-y^2[/tex]and x=7, we need to find the points of intersection between the two equations and then integrate the curve between those points.
Explanation:To find the area of the region enclosed by the graphs of [tex]x=10-y^2[/tex] and x=7, we need to find the points of intersection between the two equations. Setting x equal to each other, we have [tex]10-y^2=7.[/tex]Solving for y, we get y=±√3.
Now we can integrate the curve between the two values of y, as y goes from -√3 to √3. So the area is given by [tex]\int (10 - y^2 - 7) \, dy[/tex] from -√3 to √3.
Evaluating the integral, we get A=√3*10-2√3/3 ≈ 30.78.
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A random sample of 225 items from a population results in 60% possessing a given characteristic. Using this information, the researcher constructs a 99% confidence interval to estimate the population proportion. The resulting confidence interval is _______.
Answer: [tex](0.52,\ 0.68)[/tex]
Step-by-step explanation:
The formula for a [tex]\alpha[/tex]- level confidence interval for the population proportion:-
[tex]p\pm z_{\alpha/2}\times\sqrt{\dfrac{p(1-p)}{n}}[/tex]
Given : n = 225 ; p = 0.60 ; [tex]\alpha= 1-0.99=0.01[/tex]
By using the given information , the confidence interval for the population proportion:-
[tex]0.6\pm z_{0.005}\times\sqrt{\dfrac{0.6(1-0.6)}{225}}\\\\=0.6\pm(2.576)(0.0327)\\\\=0.6\pm0.08\\\\=0.6-0.08,\ 0.6+0.08\\\\=(0.52,\ 0.68)[/tex]
Hence, the resulting confidence interval is [tex](0.52,\ 0.68)[/tex] .
To construct a 99% confidence interval for the population proportion, we can use the sample proportion, critical value, and standard error. The resulting confidence interval is (0.516, 0.684).
Explanation:To calculate a confidence interval for a population proportion, we can use the formula:
CI = sample proportion ± (critical value)(standard error)
Given that the sample proportion is 60% (0.6) and the confidence level is 99%, we need to find the critical value corresponding to a 99% confidence level. Using a normal distribution, the critical value is approximately 2.576. The standard error can be calculated using the formula:
SE = √((sample proportion)(1 - sample proportion) / sample size)
Let's assume the sample size is 225. Plugging these values into the formula, we can calculate the standard error:
SE = √((0.6)(1 - 0.6) / 225) ≈ √(0.24 / 225) ≈ √0.001067 ≈ 0.0326
Now we can calculate the confidence interval:
CI = 0.6 ± (2.576)(0.0326) ≈ 0.6 ± 0.084
Therefore, the 99% confidence interval for the population proportion is approximately (0.516, 0.684).
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A real estate company wants to build a parking lot along the side of one of its buildings using 800 feet of fence. If the side along the building needs no fence, what are the dimensions of the largest possible parking lot?
Answer:
80,00[tex]ft^{2}[/tex]
Step-by-step explanation:
According to my research, the formula for the Area of a rectangle is the following,
[tex]A = L*W[/tex]
Where
A is the AreaL is the lengthW is the widthSince the building wall is acting as one side length of the rectangle. We are left with 1 length and 2 width sides. To maximize the Area of the parking lot we will need to equally divide the 800 ft of fencing between the Length and Width.
800 / 2 = 400ft
So We have 400 ft for the length and 400 ft for the width. Since the width has 2 sides we need to divide 60 by 2.
400/2 = 200 ft
Now we can calculate the maximum Area using the values above.
[tex]A = 400ft*200ft[/tex]
[tex]A = 80,000ft^{2}[/tex]
So the Maximum area we are able to create with 800 ft of fencing is 80,00[tex]ft^{2}[/tex]
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The largest possible parking lot, given 800 feet of fencing with one side not requiring a fence, would be a rectangular lot with dimensions of 400 feet by 200 feet.
Explanation:We're dealing with a rectangular parking lot here where one of its sides is bordered by a building, so it doesn't need a fence. We have 800 feet of fence available for the three remaining sides. Let's denote the length of the rectangular parking lot by 'x' and the width by 'y'.
Because we only need to fence three sides, we can establish the following equation based on the total amount of fence available:
x + 2y = 800
To calculate the maximum possible area of a rectangle, we need to use the formula for the area of a rectangle, which is length times width (Area = x * y). However, we want to express the area in terms of a single variable. To do this, we can rearrange our fence equation to solve for y:
y = (800 - x) / 2
Now replace y in the area equation:
Area = x * (800 - x) / 2
For the area to be maximum, the derivative of the area with respect to x must be equal to zero. Differentiating and solving for x, we get the dimensions as x = 400 and y = 200.
So the largest possible parking lot would have dimensions of 400 feet by 200 feet.
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Mr. and Mrs. Rose have six old railroad ties that they would like to use to border two different triangular flower beds. They have two ties that are 4 feet long, two ties that are 6 feet long, one tie that is 9 feet long, and one tie that is 5 feet long. Can these ties be used to border two flower beds without having to cut them? If so, what are the total possible dimensions for each set of flower beds?
Answer:
2 ties of 4 feet and 1 tie of 5 feet
2 ties of 6 feet and 1 tie of 9 feet
Step-by-step explanation:
Given data
2 ties = 4 feet
2 ties = 6 feet
1 tie = 9 feet
1 tie = 5 feet
to find out
what are the total possible dimensions for each set of flower beds
solution
there are many combination but
the best possible combination are for 1st triangular
when we use 2 ties of 4 feet and 1 tie of 5 feet
and
for 2nd triangular
when we use 2 ties of 6 feet and 1 tie of 9 feet
only these are best combination
Two surveys were done regarding credit card debt. Survey #1: Five years ago the average credit card debt was $6618. ​Survey #2:The average credit card debt for a recent year was $9205. Assume sample sizes of 35 were used and the standard deviations of both samples were $1928. Is there enough evidence to believe that the average credit card debt has changed in the past 5 years? Assume a 5% Level of Significance
Answer:
There is enough evidence to believe that the average credit card debt has changed in the past 5 years
Step-by-step explanation:
We are to compare the means of two samples. Since only sample std deviations are used, we have to use t test for this hypothesis
H0: Means are equal
Ha: Means are not equal
(Two tailed test at 5% )
Difference between means [tex]M1-M2 = -2587[/tex]
Std deviation combined = 3856
Std error for difference = 460.88
t statistic[tex]= -2587/460.88=-5.613[/tex]
p value =0
Since p <0.05 reject null hypothesis.
There is enough evidence to believe that the average credit card debt has changed in the past 5 years
This is for mathematics!p
Answer:
$933.12
Step-by-step explanation:
This is a composite figure: an upper rectangle and a lower one. We need to find the volumes of each one individually, add the volumes together, then multiply the total volume by .02
The upper rectangle has a volume of:
V = 72×12×24
V = 20,736 cubic inches
The lower rectangle has a volume of:
V = 72×36×10
V = 25,920 cubic inches
The sum of the two volumes is
V = 46,656 inches cubed
Multiply that by .02:
46,656(.02) = $933.12
Beginning one month after birth of their son, Noah, the Nelsons deposited $100 each month in an annuity for his college fund. The annuity earned interest at an average rate of 6.8% compounded monthly until his 18th birthday. What was the amount of Noah's college fund on his 18th birthday? Referring to question 4, how much interest did Noah's college fund earn in total on his 18th birthday?
Answer:
$100 * (1 + 6.8%/12)^216 + $100*(1+6.8%/12)^215 + ... + $100*(1+6.8%/12)^1
Now note that
x + x^2 + x^3 + ... + x^N = x ( 1 + x + ... + x^(N-1) )
= x ( (x^N -1)/(x-1) )
Here, x = 1+6.8%1 = 1.00566666 and N = 216, so
$100 * ( 1.00566666 ( 1.00566666^216 -1) / 0.00566666 )
= $ 42398.33
The total interest earned is $42,398 - $21,600 = $20,798
Step-by-step explanation:
The generic formula used in this compound interest calculator is V = P(1+r/n)^(nt)
V = the future value of the investment
P = the principal investment amount
r = the annual interest rate
n = the number of times that interest is compounded per year
t = the number of years the money is invested for
What would the seasons be like if the axis of Earth's rotation was tilted 0 degrees to the ecliptic, instead of the 23.5 degrees we find it today? What about if it was tilted 0 degrees?
Answer:
The seasons would become constant. It would be equinox throughout the year.
Step-by-step explanation:
The earth would be in a state of constant equinox i.e., the length of day and night would be same in a particular place.
The season of a place would be what it is when it is normally titled at equinox.
The animal and plant life which depend on the seasons would be affected.
Snow would only occur at parts where it normally snows at equinoxes.
Tickets for a play cost 2 pounds for a child, and 4 pounds for an adult. one adult brought 4 children with him and the remaining adults each brought 2 children with them. The total ticket sales were 60 pounds. how many adults and children were present in that play?
Solve using augmented matrix.
Answer:
Number of adults = 7
Number of children = 16
Step-by-step explanation:
Tickets for a play cost 2 pounds for a child, and 4 pounds for an adult.
Let x number of adults and y number of children.
1 child ticket cost = 2 pound
y children ticket cost = 2y pound
1 adult ticket cost = 4 pound
x adults ticket cost = 4x pound
Total number of ticket sales were 60 pounds
Therefore, 4x + 2y = 60 ------------- (1)
One adult brought 4 children with him and the remaining adults each brought 2 children with them.
Remaining number of adult whose brought 2 children = x-1
Number children = 2(x-1)
Total number of children = 2(x-1)+4
Therefore, y=2x+2 ---------------------(2)
System of equation,
2x + y = 30
-2x + y = 2
Using augmented matrix to solve system of equation.
[tex]\begin{bmatrix}2&1&\ |30\\-2&1&|2\end{bmatrix}\\\\R_2\rightarrow R_2+R_1\\\\\begin{bmatrix}2&1& |30\\0&2&|32\end{bmatrix}\\\\R_2\rightarrow\dfrac{1}{2}R_2\\[/tex]
[tex]\begin{bmatrix}2&1&\ |30\\0&1&|16\end{bmatrix}\\\\R_1\rightarrow R_1-R_2\\\\\begin{bmatrix}2&0&\ |14\\0&1&|16\end{bmatrix}\\\\\\[/tex]
[tex]R_1\rightarrow \dfrac{1}{2}R_1\\\\\begin{bmatrix}1&0&|7\\0&1&|16\end{bmatrix}\\\\[/tex]
Now, we find the value of variable.
[tex]x=7\text{ and }y=16[/tex]
Hence, Number of adults are 7 and Number of children are 16.
A company that manufactures and bottles apple juice uses a machine that automatically fills 16–ounce bottles. There is some variation in the amounts of liquid dispensed into the bottles. The amount dispensed is approximately normally distributed with mean 16 ounces and standard deviation 1 ounce. What proportion of bottles will have more than 17 ounces?
Answer: 0.1587
Step-by-step explanation:
Given : The amount dispensed is approximately normally distributed with Mean : [tex]\mu=\ 16[/tex]
Standard deviation : [tex]\sigma= 1[/tex]
The formula to calculate the z-score :-
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x= 17
[tex]z=\dfrac{17-16}{1}=1[/tex]
The p-value =[tex] P(17<x)=P(1<z)[/tex]
[tex]=1-P(z<1)=1-0.8413447\\\\=0.1586553\approx0.1587[/tex]
The proportion of bottles will have more than 17 ounces = 0.1587
Write the following equation in slope-intercept form and identify the slope and y-intercept 5x+3y 15 The equation in slope-intercept form is y (Simplify your answ er. Type your answer in slope-intercept form. Use integers or fractions for any numbers in the expression.)
To write the equation 5x + 3y = 15 in slope-intercept form, solve for y to get y = (-5/3)x + 5. The slope is -5/3 and the y-intercept is 5.
To convert the equation 5x + 3y = 15 into slope-intercept form, which is y = mx + b, we need to solve for y. Here are the steps:
Subtract 5x from both sides: 3y = -5x + 15.Divide every term by 3 to isolate y: y = (-5/3)x + 5.In this slope-intercept form, the coefficient of x represents the slope and the constant term represents the y-intercept. Therefore, the slope is -5/3 and the y-intercept is 5. This tells us that for every increase of 1 on the horizontal axis (x), there is a decrease of 5/3 on the vertical axis (y), and the line crosses the y-axis at the point (0, 5).How many dfferent strings can be made from the letters in MISSISSIPPI, using all letters? 15 013837
Answer: 34650
Step-by-step explanation:
The number of permutations of n objects, where one object is repeated [tex]n_1[/tex] times , another is repeated [tex]n_2[/tex] times and so on is :
[tex]\dfrac{n!}{n_1!n_2!....n_k!}[/tex]
Given : The number of letters in string MISSISSIPPI = 11
Here I is repeated 4 times, S is repeated 4 times and P is repeated 2 times.
Then , the number of different strings can be made from the letters in MISSISSIPPI, using all letters is given by :-
[tex]\dfrac{11!}{4!4!2!}=34650[/tex]
Therefore , there are 34650 different strings can be made from the letters in MISSISSIPPI.
To find the number of different strings that can be formed from the letters in MISSISSIPPI, we use the formula for permutations of a multi set. The total is 34,650 unique permutations.
The question asks how many different strings can be made from the letters in MISSISSIPPI, using all letters.
This is a problem of calculating permutations of a multi set.
The word MISSISSIPPI has 11 letters with the following counts of each letter: M occurs 1 time, I occurs 4 times, S occurs 4 times, and P occurs 2 times.
To find the total number of unique permutations, we use the formula for the permutations of a multiset:
The formula is:
Permutations = n! / (n1! * n2! * ... * nk!),
where n is the total number of items to arrange, and n1, n2, ..., nk are the counts of each distinct item.
For MISSISSIPPI:
Total permutations = 11! / (1! * 4! * 4! * 2!) = 34650
Therefore, there are 34,650 different strings that can be made from the letters in MISSISSIPPI using all the letters.
Write the equation in the slope-intercept form 7x +4y - 20 0 Find the y-intercept of the corresponding line. (x, y)
Step-by-step explanation:
write the equation in slope intercept form
7x + 4y - 20=0
find the slope of corresponding line.
????
then
find the y intercept of corresponding line
(x,y)=???.
2. Let A be a 3 x 3 matrix such that det(A)= -4. Find det(3A)
Answer:
The value of det (3A) is -108.
Step-by-step explanation:
If M is square matrix of order n x n, then
[tex]|kA|=k^n|A|[/tex]
Let as consider a matrix A or order 3 x 3. Using the above mentioned property of determinant we get
[tex]|kA|=k^3|A|[/tex]
We need to find the value of det(3A).
[tex]|3A|=3^3|A|[/tex]
[tex]|3A|=27|A|[/tex]
It is given that the det(A)= -4. Substitute |A|=-4 in the above equation.
[tex]|3A|=27(-4)[/tex]
[tex]|3A|=-108[/tex]
Therefore the value of det (3A) is -108.
Find 10^(5^101) (mod 21).
note: 10^(5^101) is not 10^(501)
We have [tex]\lambda(21)=6[/tex], where [tex]\lambda[/tex] is the Carmichael function. So we have
[tex]10^{5^{101}}\equiv10^{5^{101}\pmod6}\pmod{21}[/tex]
The powers of 5 modulo 6 follow a periodic pattern
[tex]5^1\equiv5\pmod6[/tex]
[tex]5^2\equiv25\equiv1\pmod6[/tex]
[tex]5^3\equiv1\cdot5\equiv5\pmod6[/tex]
[tex]5^4\equiv5^2\equiv1\pmod6[/tex]
and so on, with odd powers of 5 equivalent to 5 modulo 6. So
[tex]10^{5^{101}}\equiv10^{5^{101}\pmod6}\equiv10^5\pmod{21}[/tex]
The rest is easy to deal with. We have
[tex]10^2\equiv16\pmod{21}[/tex]
[tex]10^3\equiv160\equiv13\pmod{21}[/tex]
[tex]10^4\equiv130\equiv4\pmod{21}[/tex]
[tex]10^5\equiv40\equiv19\pmod{21}[/tex]
and so the answer is 19.
Use the given data to find the 95% confidence interval estimate of the population mean μ. Assume that the population has a normal distribution. IQ scores of professional athletes: Sample size n=10 Mean x¯=104 Standard deviation s=10
With a 95% confidence level, the population mean is estimated to be between approximately 96.85 and 111.15 based on a sample size of 10, a mean of 104, and a standard deviation of 10.
With a sample size (n) of 10, a mean \bar{x}104, and a standard deviation (s) of 10, we can find the 95% confidence interval for the population mean (μ).
First, we calculate the standard error of the mean (SE). The standard error of the mean can be calculated by dividing the standard deviation by the square root of the sample size.
SE = s/√n.
By substituting s = 10 and n = 10 into the equation, we get SE = 3.162277660168379.
Next, we need to find the critical value (t) for a 95% confidence interval based on a t-distribution. Since we're using a confidence level of 95% and the sample size is 10, which means degree of freedom is n-1=9, the critical value (t) is 2.2621571627409915 based on the t-distribution table.
To calculate the lower bound and the upper bound of the 95% confidence interval, you should subtract and add to the mean the product of the critical value and the standard error respectively.
So,
Lower Bound = \bar{x} - t * SE
Upper Bound = \bar{x} + t * SE
Substituting from our known values, we get:
Lower Bound = 104 - 2.2621571627409915 * 3.162277660168379 = 96.84643094047428
Upper Bound = 104 + 2.2621571627409915 * 3.162277660168379 = 111.15356905952572
So, with a 95% confidence level, the confidence interval estimate of the population mean is (96.84643094047428, 111.15356905952572). This means we are 95% confident that the true population mean lies somewhere between approximately 96.85 and 111.15.
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The 95% confidence interval for the population mean IQ score of professional athletes, based on a sample size of 10 with a mean of 104 and standard deviation of 10, is estimated to be between 96.83 and 111.17.
To find the 95% confidence interval estimate of the population mean [tex](\( \mu \))[/tex] given the sample data, we'll use the formula for the confidence interval for a population mean when the population standard deviation is unknown:
[tex]\[ \text{Confidence interval} = \bar{x} \pm t \left( \frac{s}{\sqrt{n}} \right) \][/tex]
Where:
-[tex]\( \bar{x} \)[/tex] is the sample mean,
- s is the sample standard deviation,
- n is the sample size, and
- t is the critical value from the t-distribution for the desired confidence level and degrees of freedom.
Given:
- Sample size n = 10
- Sample mean [tex](\( \bar{x} \))[/tex]= 104
- Sample standard deviation s = 10
First, we need to find the critical value t for a 95% confidence level with 9 degrees of freedom (since n - 1 = 10 - 1 = 9 ).
Using a t-table or statistical software, [tex]\( t \approx 2.262 \)[/tex] for a 95% confidence level and 9 degrees of freedom.
Now, let's plug in the values into the formula:
[tex]\[ \text{Confidence interval} = 104 \pm 2.262 \left( \frac{10}{\sqrt{10}} \right) \][/tex]
Now, let's calculate the margin of error:
[tex]\[ \text{Margin of error} = 2.262 \left( \frac{10}{\sqrt{10}} \right) \]\[ \text{Margin of error} \approx 7.17 \][/tex]
Finally, let's calculate the confidence interval:
[tex]\[ \text{Lower bound} = 104 - 7.17 \]\[ \text{Upper bound} = 104 + 7.17 \]\[ \text{Lower bound} \approx 96.83 \]\[ \text{Upper bound} \approx 111.17 \][/tex]
So, the 95% confidence interval estimate of the population mean IQ score of professional athletes is approximately between 96.83 and 111.17.
Alfred collects black and brown socks but loses a lot of them. If 60% of all his socks are brown, 20% of his brown socks are in the wash and 120 brown socks are in his sock drawer, how many socks does Alfred have?
Answer:
Alfred has 250 socks in his collection.
Step-by-step explanation:
Hello, great question. These types are questions are the beginning steps for learning more advanced Algebraic Equations.
Based on the information given to us we can see that out of All his Brown socks he has 20% in the wash and the rest are in his drawer. Meaning 80% of the Brown socks are in his drawer. So we first need to find how many Brown socks are in the wash. We can solve this using the Rule of Three property as shown in the picture below.
120 drawer ⇒ 80%
x wash ⇒ 20%
[tex]\frac{120*20}{80} = 30[/tex]
Now that we have the amount of Brown socks in the washer we can add that to the amount in the drawer to find the total amount of Brown socks.
[tex]Br = 120+30\\Br = 150[/tex]
So we now know that there are a total of 150 Brown socks. Since the question states that the Brown socks are 60% of the total we can use the Rule of Three to find the total.
150 Brown ⇒ 60%
T Total ⇒ 100%
[tex]\frac{150*100}{60} = T[/tex]
[tex]250 socks = T[/tex]
Finally, we can see that Alfred has 250 socks in his collection.
I hope this answered your question. If you have any more questions feel free to ask away at Brainly.
Alfred owns a total of 250 socks. This was derived by first finding out that Alfred has 150 brown socks, which represent 60% of his total socks' collection. Thus, the total number of socks Alfred owns is 250.
Explanation:The question deals with the calculation of a total number of socks owned by Alfred.
If we know that 120 brown socks represent 80% of all of Alfred's brown socks (because 20% of them are in the wash), we can calculate the total number of brown socks. To do this, divide 120 by 0.8, which equals 150. Hence, Alfred has 150 brown socks.
We know from the problem that the brown socks account for 60% of all his socks. Hence the total number of socks (brown and black) is calculated by dividing the total number of brown socks (150) by 0.6. After performing this percentage calculation, we find that Alfred owns 250 socks.
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8) The monthly worldwide average number of airplane crashes of commercial airlines is 3.5. What is the probability that there will be (a) at least 2 such accidents in the next month; (b) at most 1 accident in the next month? Explain your reasoning!
Answer: (a) 0.8641
(b) 0.1359
Step-by-step explanation:
Given : The monthly worldwide average number of airplane crashes of commercial airlines [tex]\lambda= 3.5[/tex]
We use the Poisson distribution for the given situation.
The Poisson distribution formula for probability is given by :-
[tex]P(X=x)=\dfrac{e^{-\lambda}\lambda^x}{x!}[/tex]
a) The probability that there will be at least 2 such accidents in the next month is given by :-
[tex]P(X\geq2)=1-(P(X=1)+P(X=0))\\\\=1-(\dfrac{e^{-3.5}(3.5)^0}{0!}+\dfrac{e^{-3.5}(3.5)^1}{1!})\\\\=1-(0.1358882254)=0.8641117746\approx0.8641[/tex]
b) The probability that there will be at most 1 accident in the next month is given by :-
[tex]P(X\leq1)=(P(X=1)+P(X=0))\\\\=\dfrac{e^{-3.5}(3.5)^0}{0!}+\dfrac{e^{-3.5}(3.5)^1}{1!}\\\\=0.1358882254\approx0.1359[/tex]
If a population is recorded at 1,200 in the year 2000 and the rate of increase is a steady 50 people each year, what will be the population in 2018?
Answer:
2100
Step-by-step explanation:
50*18=900
900+1,200=2100
To find the population in 2018, we calculate the total increase from 2000 to 2018 by multiplying the yearly increase of 50 people by 18 years, resulting in an additional 900 people. Adding this to the initial population of 1,200 people gives us a total population of 2,100 people in 2018.
If a population is recorded at 1,200 in the year 2000 and increases at a steady rate of 50 people each year, we can calculate the population in 2018 using a linear growth model. First, we need to determine the number of years between 2000 and 2018, which is 18 years. Next, we multiply the annual increase (50 people) by the number of years (18) to find the total increase over this period.
The calculation would be as follows:
Total Increase = Annual Increase times Number of YearsTotal Increase = 50 people/year times 18 yearsTotal Increase = 900 peopleWe then add this total increase to the initial population to get the population in 2018:
Population in 2018 = Initial Population + Total IncreasePopulation in 2018 = 1,200 people + 900 peoplePopulation in 2018 = 2,100 peopleThe population in 2018 would be 2,100 people.
In the past, 35% of the students at ABC University were in the Business College, 35% of the students were in the Liberal Arts College, and 30% of the students were in the Education College. To see whether or not the proportions have changed, a sample of 300 students from the university was taken. Ninety of the sample students are in the Business College, 120 are in the Liberal Arts College, and 90 are in the Education College. The expected frequency for the Business College is
Answer:
Step-by-step explanation:
Given that in the past, 35% of the students at ABC University were in the Business College, 35% of the students were in the Liberal Arts College, and 30% of the students were in the Education College.
Table is prepared as follows
Bus coll. Lib Arts coll Educ. coll Total
Observed 90 120 90 300
Expected p.c. 35 35 30 100
Expected number 105 105 90 300
Percent*300/100
Hence expected frequency for business college = 105
A bag of 100 tulip bulbs purchased from a nursery contains 20 red tulip bulbs, 20 yellow tulip bulbs, and 60 purple tulip bulbs. (a) What is the probability that a randomly selected tulip bulb is red? (b) What is the probability that a randomly selected tulip bulb is purple? (c) Interpret these two probabilities.
Answer: a) 0.2 b) 0.6
c) The event of selecting red tulip is not likely to occur.
The event of selecting purple tulip is likely to occur.
Step-by-step explanation:
Given : Total number of tulips = 100
The number of red tulips = 20
The number of purple tulips =60
The probability that a randomly selected tulip bulb is red :-
[tex]\dfrac{\text{Number of red tulips}}{\text{Total tulips}}\\\\=\dfrac{20}{100}=0.2[/tex]
Since 0.2 is less than 0.5.
It means that the event of selecting red tulip is not likely to occur.
The probability that a randomly selected tulip bulb is purple :-
[tex]\dfrac{\text{Number of purple tulips}}{\text{Total tulips}}\\\\=\dfrac{60}{100}=0.6[/tex]
Since 0.6 is more than 0.5.
It means that the event of selecting purple tulip is likely to occur.
Final answer:
The probability of selecting a red tulip bulb is 20%, and the probability of selecting a purple tulip bulb is 60%. These probabilities reflect the likelihood of picking a bulb of a particular color at random from the bag.
Explanation:
The question involves calculating the probability of selecting a red or purple tulip bulb from a bag.
Probability of Selecting a Red Tulip Bulb
The probability, P(Red), is calculated by dividing the number of red bulbs by the total number of bulbs:
P(Red) = Number of Red Bulbs / Total Number of Bulbs = 20 / 100 = 0.2
Probability of Selecting a Purple Tulip Bulb
Similarly, the probability, P(Purple), is:
P(Purple) = Number of Purple Bulbs / Total Number of Bulbs = 60 / 100 = 0.6
Interpretation of Probabilities
These probabilities indicate that there is a 20% chance of selecting a red bulb and a 60% chance of selecting a purple bulb from the bag. The higher the probability, the more likely it is to select a bulb of that color at random.
(5). (10 points) There are 5 hotels in Stony Brook. If 3 people check into hotels on September 12, what is the probability that they each check into a different hotel? (What assumptions are you making?) Make sure to define any notation you use to describe elements of the sample space.
Answer:
0.48
Step-by-step explanation:
Probability that the first person chooses a hotel
⁵C₁
[tex]^5C_1=\frac{5!}{(5-1)!1!}\\=\frac{120}{24}=5[/tex]
Probability that the second person chooses a different hotel
⁴C₁
[tex]^4C_1=\frac{4!}{(4-1)!1!}\\=\frac{24}{6}=4[/tex]
because the choice of hotels has reduced by 1 as one hotel is occupied by the first person
Probability that the second person chooses a different hotel
³C₁
[tex]^3C_1=\frac{3!}{(3-1)!1!}\\=\frac{6}{2}=3[/tex]
because the choice of hotels has reduced by 2 as two different hotels are occupied by the first person and second person
∴ The favorable outcomes are =⁵C₁×⁴C₁׳C₁=5×4×3=60
The total number of outcomes=5³=125
∴Probability that they each check into a different hotel=60/125=0.48