A rectangular prism has the following dimensions: l = 5a , w = 2a ,
h = ( a^3 - 3a^2 + a ) Use the formula V = l ⋅ w ⋅ h to find the volume of the rectangular prism.
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The volume of a shape is the amount of space in it.
The volume of the rectangular prism is: [tex]\mathbf{10a^5 -30a^4 + 10a^3}[/tex]
The dimensions of the rectangular prism are:
[tex]\mathbf{l = 5a}[/tex]
[tex]\mathbf{w = 2a}[/tex]
[tex]\mathbf{h = (a^3 - 3a^2 + a)}[/tex]
The volume (v) of the rectangular prism is:
[tex]\mathbf{v = l\cdot w \cdot h}[/tex]
So, we have:
[tex]\mathbf{v = 5a \cdot 2a \cdot (a^3 -3a^2 + a)}[/tex]
[tex]\mathbf{v = 10a^2 \cdot (a^3 -3a^2 + a)}[/tex]
Expand
[tex]\mathbf{v = 10a^5 -30a^4 + 10a^3}[/tex]
Hence, the volume of the rectangular prism is: [tex]\mathbf{10a^5 -30a^4 + 10a^3}[/tex]
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G for exercises 13 and 14, assume the sample populations do not have equal standard deviations and use the .05 significance level: (a) determine the number of degrees of freedom, (b) state the decision rule, (c) compute the value of the test statistic, and (d) state your decision about the null hypothesis.
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John's goal is to have more than -7 dollars in his bank account by the end of the month. The variable d is the number of dollars in John's bank account at the end of the month. Write an inequality in terms of d that is true only if John meets his monthly goal.
Answers
Answer:
d > - 7
Step-by-step explanation:
Here, d represents the number of dollars in John's bank account at the end of the month.
As per statement,
In John's account,
The balance must be more than -7 dollars,
That is, Number of dollars at the end of each month > - 7 ( '>' is used for more than )
⇒ d > - 7
Which is the required inequality.
An open rectangular box has volume 32 cm3. find the lengths of the edges giving the minimum surface area
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An open rectangular box with a volume of 32 cm³ would have its minimum surface area when it takes the form of a cube. In this case, each side of the box would be approximately 3.174 cm.
Explanation:To minimize the surface area of an open rectangular box given a constant volume, the box should take the shape of a cube. This is because a cube is the three-dimensional shape that has the minimum surface area for a given volume. The volume V of the box is given as 32 cm³. Since for a cube, the volume V = a³ where a is the side length of the cube, we need to find the value of a such that the volume is equal to 32cm³.
To do this, we use the cube root function as follows: a = ∛V = ∛32 = 3.174 cm.
So, for a rectangular box with a volume of 32 cm³, the lengths of the edges that would give the minimum surface area would all be approximately 3.174 cm.
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An open rectangular box with a volume of 32 cm³ would have its minimum surface area when it takes the form of a cube.
In this case, each side of the box would be approximately 3.174 cm.
To minimize the surface area of an open rectangular box given a constant volume, the box should take the shape of a cube.
This is because a cube is the 3-dimensional shape that has the minimum surface area for a given volume.
The volume V of the box is given as 32 cm³.
Since for a cube, the volume V = a³ where a is the side length of the cube, we need to find the value of a such that the volume is equal to 32cm³.
To do this, we use the cube root function as follows:
a = ∛V
= ∛32
= 3.174 cm.
So, for a rectangular box with a volume of 32 cm³, the lengths of the edges that would give the minimum surface area would all be approximately 3.174 cm.
if f(x)=x-3/x, g(x)=x+3, and h(x)=2x+1, what is (g*h*f)(x).
A. 3x-6/x
B. 6x-6/x
C. 2x+4/2x+7
D. 4x-3/x
Answers
Answer:
Option B - 6x-6/x
Step-by-step explanation:
Given : If [tex]f(x)=\frac{x-3}{x}, g(x)=x+3,h(x)=2x+1[/tex].
To find : What is the value of [tex](g(h(f)))(x)[/tex]
Solution :
[tex](g(h(f)))(x)=g(h(f(x)))[/tex]
i.e, substitute the value of f(x) in h(x) and then put that h(x) in g(x).
[tex]g(h(f(x)))=g(h(\frac{x-3}{x}))[/tex]
[tex]g(h(f(x)))=g(2(\frac{x-3}{x})+1)[/tex]
[tex]g(h(f(x)))=(2(\frac{x-3}{x})+1)+3[/tex]
Solving the RHS,
[tex]=(2(\frac{x-3}{x})+1)+3[/tex]
[tex]=(\frac{2x-6}{x})+1)+3[/tex]
[tex]=(\frac{2x-6+x}{x})+3[/tex]
[tex]=\frac{3x-6}{x}+3[/tex]
[tex]=\frac{3x-6+3x}{x}[/tex]
[tex]=\frac{6x-6}{x}[/tex]
Therefore, The value of [tex](g(h(f)))(x)=\frac{6x-6}{x}[/tex]
Hence, Option B is correct.
The data set gives the number of hours it took each of the 10 students in a cooking class to master a particular technique. Which of the numbers below is the best measure of the central tendency of the data? {3, 3, 4, 4, 4, 5, 5, 5, 5, 30}
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Given line L = ax + by + c = 0, b(its suppose to be the equal sign with the slash going thru it) 0
What is the slope of this function?
-a/b
a/b
b/a
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by = - ax - c
y = -a/b (x) - c/b
so slope = -a/b
answer
-a/b
Is this less than
Greater than
Or equal too?
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Root 5.76= 2.4
-2.4=-2.4
can someone please explain proportions?
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A statement or assertion that exppresses a judment or opinion
In the x -y plane, the line 3x + 2y =72 intersects the x -axis at x=b,for some number b. What is the value of b?
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Since it is specifically stated that the line intersects the x-axis, then this means that y = 0. Therefore substituting y = 0 into the given equation will us the value of x.
Substituting y = 0:
3 x + 2 * 0 = 72
3 x = 72
x = 24
Therefore this means that the line intersect the x-axis at point (24, 0).
You and two friends (Adam and Alana) will only play a game if it is fair for all three of you. The game your friend has proposed is to roll two number cubes (numbered 1-6) and find the sum. If the sum. If the sum is from 1-4 you get the point, 5-8 Adam gets the point, and 9-12 Alana gets the point.
Answers
Yes, the game was fair and no one had a better advantage.
Answer:
yes!
Step-by-step explanation:
no one had a better advantage.everyone had the same amount of advantage.
The product of (n^2-6n+3)and -4 is
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-4n^2+24n-12
making use of Archimedes’s Principle, displacements, and buoyancy, explain why some objects float on water, but others do not.
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Then, there are two forces acting on the body, its weight (vertical downwards) and the bouyancy (vertical push upwards).
There are three possibilities:
1) The buoyancy is greater than the weight of the body => the body will float (move upward, toward the surface if it is a liquid)
2) The buoyancy is equal than the weight of the boy => the body wll remain quite (also floating but not moving either upwards or downwards)
3) The buoyancy is less than the weight of the body => the body will sinkl.
So, a body will float on water when the buoyancy from the liquid is overcomes its weight.
Buoyancy is related with density because:
buoyancy = weight of the liquid displaced = mass o fliquid * g = density of the liquid * Volume of the liquid * g
Weight of the body = mass of the body * g = density of the body * Volumen of the body * g
When the body is completely immersed in the liquid its volume es equal to the volume of the liquid displaced =>
So we can compare the weight of the body and the buouancy force
density of liquid * volume of liquid * g vs density of body * volume of liquid * g
where the difference is the densities of liquid and body.
That is why it is deduced that the bodies float when their densities are smaller than the densities of the liquid where they are.
Answer:
If the weight is equal to the buoyant force, it will float. The same is true for if the weight is less than the buoyant force. However, if the weight is greater than the buoyant force, it will sink.
How many different three digit numbers can be written using digits 1,2,3,4,5 without any repeating digits? Thank you.
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The 1st digit can be written in 5 ways
The 2nd digit can be written in 4 ways (since 1 already gone and no repetition)
The 3rd digit can be written in 3 ways (since 2 already gone and no repetition)
Total number of ways = 5x4x3 = 60 ways
2nd Method : Permutation (since ORDER MATTERS) of 3 digits chosen among 5 (with no repetition)
⁵P₃ = (5!)/(5-3)! = (5!)/(2!) = 60 ways
We can use the principles of permutation to find out that there are 60 different three-digit numbers that could be formed using the digits 1,2,3,4,5 without repetition.
Explanation:In order to find out how many different three-digit numbers can be formed using the digits 1,2,3,4,5 without repetition, we use the principles of permutation. Firstly, we should determine how many options we have for each digit place in the three-digit number.
There are 5 possible digits for the first place (since none has been used yet).There are 4 remaining digits for the second place.Finally, there are only 3 digits left to fill the third place.We then multiply these options together (5 x 4 x 3) which equals to 60 different three-digit numbers.
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PLEASE HELP?? EASY 5 POINTS!!! The two models shown have the same volume. Complete the equation and expression below about the volume of each figure.
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36 in³ + 12 in³ = 48 in³
Volume of the Blue Prism in Model 1:
48 in³ - 16 in³ = 32 in³
Write a function rule for the table. (2, $16.00), (4, $32.00), (6, $48.00), (8, $64.00)
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Evaluate the line integral where f(x,y,z)=3sin(x)i−(cos(y))j−5xzk and c is given by the vector function r(t)=t5i−t4j+t3kr(t)=t5i−t4j+t3k , 0≤t≤10≤t≤1.
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[tex]\mathbf r(t)=\mathbf r'(t)\,\mathrm dt=t^5\,\mathbf i-t^4\,\mathbf j+t^3\,\mathbf k[/tex]
[tex]\implies\mathrm d\mathbf r(t)=(5t^4\,\mathbf i-4t^3\,\mathbf j+3t^2\,\mathbf k)\,\mathrm dt[/tex]
[tex]\displaystyle\int_C\mathbf f(x,y,z)\cdot\mathrm d\mathbf r=\int_{t=0}^{t=1}f(x(t),y(t),z(t))\cdot\mathbf r'(t)\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^1(3\sin(t^5)\,\mathbf i-\cos(t^4)\,\mathbf j-5t^8\,\mathbf k)\cdot(5t^4\,\mathbf i-4t^3\,\mathbf j+3t^2\,\mathbf k)\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^1(15t^4\sin(t^5)+4t^3\cos(t^4)-15t^{10})\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^1\bigg(3\sin(t^5)\,\mathrm d(t^5)+\cos(t^4)\,\mathrm d(t^4)-15t^{10}\,\mathrm dt\bigg)[/tex]
[tex]=-3\cos(t^5)+\sin(t^4)-\dfrac{15}{11}t^{11}\bigg|_{t=0}^{t=1}[/tex]
[tex]=-3\cos1+\sin1+\dfrac{18}{11}[/tex]
Write out the form of the partial fraction decomposition of the function (see example). do not determine the numerical values of the coefficients. (3x + 1) / (x + 1)^3(x^2 + 8)^2
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can be expanded as
[tex]\dfrac a{x+1}+\dfrac b{(x+1)^2}+\dfrac c{(x+1)^3}+\dfrac{dx+e}{x^2+8}+\dfrac{fx+g}{(x^2+8)^2}[/tex]
Lori rented a car for one day. The daily rental rate was $12.50 plus 25 cents per mile. The total cost of her rental was $46. How many miles did she drive that day?
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33.5=.25x
x=134
Final answer: 134 miles
A 20-year, $148,000 mortgage is taken out with a 4.9% APR. If payments are scheduled monthly and there are no additional costs, determine the balance on the account after the first month's payment has been made.
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[tex]\bf \qquad \qquad \textit{Amortized Loan Value} \\\\ pymt=P\left[ \cfrac{\frac{r}{n}}{1-\left( 1+ \frac{r}{n}\right)^{-nt}} \right][/tex]
[tex]\bf \qquad \begin{cases} P= \begin{array}{llll} \textit{original amount deposited}\\ \end{array}\to & \begin{array}{llll} 148,000 \end{array}\\ pymt=\textit{periodic payments}\\ r=rate\to 4.9\%\to \frac{4.9}{100}\to &0.049\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{payments are monthly, thus} \end{array}\to &12\\ t=years\to &20 \end{cases} \\\\\\ pymt=148000\left[ \cfrac{\frac{0.049}{12}}{1-\left( 1+ \frac{0.049}{12}\right)^{-12\cdot 20}} \right][/tex]
How would you use the Fundamental Theorem of Calculus to determine the value(s) of b if the area under the graph g(x)=4x between x=1 and x=b is equal to 240?
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[tex] \int\limits^b_a g{(x)} \, dx = F(b)-F(a)[/tex]
Where F is the indefinite integral of [tex]g(x)[/tex]
The first step is to integrate [tex]g(x)[/tex]
[tex] \int\ {4x} \, dx = \frac{4x^{1+1} }{1+1} = \frac{4x^{2} }{2} =2 x^{2} [/tex]
Then substitute the value of [tex]b[/tex] and [tex]a=1[/tex] into [tex] 2x^{2} [/tex]
[tex][2 (b)^{2}]-[2 (1)^{2}] = 240 [/tex]
[tex] 2b^{2} -2=240[/tex]
[tex] 2b^{2}=240+2 [/tex]
[tex] 2b^{2}=242 [/tex]
[tex] b^{2}= \frac{242}{2} [/tex]
[tex] b^{2}=121 [/tex]
[tex]b=11[/tex]
Hence the limit of the area under [tex]g(x)[/tex] is between [tex]a=1[/tex] and [tex]b=11[/tex]
Solve the equation by completing the square. Round to the nearest hundredth if necessary. x2 – x = 25
Answers
Add c in both side
c = (b/2a)^2 = (1/2)^2 = 1/4
x^2 - x + 1/4 = 25 + 1/4
(x - 1/2)^2 = 101/4
(x - 1/2) = √101/4
x - 1/2 = 5.02
x = 5.52 &
x = 4.52
5.52, –4.52
is the correct answer,never forget negatives!
Simplify 3 - (2 - 8).
Answers
3 - ( -6 ) =
3 + 6 =
9
If f(x) = 5x – 6, what is the equation for f–1(x)?
Answers
Answer:
5x-2+^3 hole
Step-by-step explanation:
The inverse of f(X) = (x +6 ) /5
What is inverse function?The inverse of a function f is denoted by f^-1 and it exists only when f is both one-one and onto function. Note that f^-1 is NOT the reciprocal of f. The composition of the function f and the reciprocal function f-1 gives the domain value of x.
(f o f^-1) (x) = (f^-1 o f) (x) = x
Steps To Find An Inverse FunctionThe following sequence of steps would help in conveniently finding the inverse of a function. Here we consider a function f(x) = ax + b, and aim at finding the inverse of this function through the following steps.
For the given function f(x) = ax + b, replace f(x) = y, to obtain y = ax + b.Interchange the x with y and the y with x in the function y = ax + b to obtain x = ay + b.Here solve the expression x = ay + b for y. And we obtain y = (x - b/aFinally replace y = f^-1(x), and we have f^-1(x) = (x - b)/a.Given:
f(x) = 5x – 6
let y= f(x) = 5x – 6
Now, Interchange the x with y and the y with x.
x= 5y -6
x+6 = 5y
y = (x +6 ) /5
So, f^-1(x) = (x +6 ) /5
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The length of a spring is a linear function of the weight of the object compressing it. For a particular spring, the length decreases by 1.5 inches for every 3 pounds pressing down on it. When 4 pounds are on the spring, the length of the spring is 8 inches long. What is the length of the spring if 15 pounds are pressing down on the spring?
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EASY 5 POINTS!!! You have a new pool and want to know its volume. The pool is 5 feet deep and has a radius of 7 feet. About how much water can the pool hold? Use 3.14 for pi.
Answers
Answer:
The volume of water that can be hold by the pool is approximately 770 cubic feet.
Step-by-step explanation:
To calculate it we have to use the volume formula for a cylinder. This is:
[tex]Volume=pi*r^{2}*h\\pi=3.14\\r=radius\\h=deep[/tex]
Solving the equation with the given values:
[tex]Volume=3.14*7^{2} *5\\Volume=769.3 cubic feet \\Aprox.Volume= 770 cubic feet[/tex]
The volume of water that can be hold by the pool is approximately 770 cubic feet.
The number of bagels sold daily for two bakeries is shown in the table. Bakery A Bakery B 53 34 52 40 50 36 48 38 53 41 47 44 55 40 51 39 Based on these data, is it better to describe the centers of distribution in terms of the mean or the median? Why? Select the correct answer below.
Answers
bakery A
mean (A) = [53 + 34 + 52 + 40 + 50 + 36 + 48 + 38]/8 = 43.88
Bakery B
mean (B) = [53 + 41 + 47 + 44 + 55 + 40 + 51 + 39]/8 = 46.25
Even if bakery A and B has the 38 and 39 as the lowest data respectively, they are still near the average data.
Considering the information about the asymmetry in Bakery B's distribution, the median would be a more appropriate measure of central tendency for both Bakery A and B. It would provide a better representation of the "typical" number of cupcakes sold in each bakery, even if there might be outliers in the data.
Understanding Mean and Median:
Mean: The mean, also known as the average, is calculated by adding all the values in a dataset and dividing by the number of values. It's a good measure of central tendency when the data is symmetrical and free from outliers.Median: The median is the 'middle' value when the data is ordered from least to greatest. It's less sensitive to extreme values (outliers) compared to the mean.Why Median is More Appropriate (Considering the Information):
The information mentions that the distribution for Bakery B is not symmetric. This suggests that the data points in Bakery B might be skewed towards one side, with a few possibly higher or lower values compared to the majority.
Reasoning:
Option A (Mean): The mean is easily influenced by outliers. If Bakery B has outliers, the mean might not accurately represent the "typical" number of cupcakes sold for Bakery B.Option B (Mean for Perfect Symmetry): This option specifies the mean if the distributions are perfectly symmetrical. Since we know Bakery B's distribution isn't symmetrical, the mean wouldn't be the best choice.Option C (Median for Bakery B Asymmetry): This correctly identifies that the median is a better choice for Bakery B due to the asymmetry in its data.Option D (Median for Outliers): While outliers can be present in both Bakery A and B, the key factor is the asymmetry in Bakery B's distribution. The median is generally more resistant to the influence of outliers, making it a better choice in this case, even if outliers exist in both datasets.complete question:
The number of bagels sold daily for two bakeries is shown in the following table: Uchange Bakery A Bakery B 53 34 52 40 50 36 48 38 53 41 47 44 55 40 51 39 Based on these data, is it better to describe the centers of distribution in terms of the mean or the median? Why? Select the correct answer below. (3 points)
a. Mean, because there are no outliers to affect the data
b. Mean, because the distribution is perfectly symmetric for both bakeries
c. Median, because the distribution for Bakery B is not symmetric
d. Median, because there are outliers that affect both data sets
2/3 (5 x 4) ÷ 3 + 6/7 I divided 3/2 is 1.5*(20)=30/3=10+1.16=11.16 is that correct not so good in math
Answers
2/3 (5 × 4) ÷ 3 + 6/7 = 334/63
Further explanationOrder of Operations in Mathematics follow this following rule :
ParenthesesExponentsMultiplication and DivisionAddition and SubtractionThis rule is known as the PEMDAS method.
In working on a mathematical problem, we first calculate operation that is in parentheses, follow by exponentiation, then multiplication or division, and finally addition or subtraction.
Let us tackle the problem.
Let's write down the problem that is wanted to be solved:
[tex]\frac{2}{3} ( 5 \times 4 ) \div 3 + \frac{6}{7}[/tex]
[tex]\frac{2}{3} ( 20 ) \div 3 + \frac{6}{7}[/tex] → Parentheses First
[tex]\frac{40}{3} \div 3 + \frac{6}{7}[/tex] → Multiplication
[tex]\frac{40}{9} + \frac{6}{7}[/tex] → Division
[tex]\frac{280}{63} + \frac{54}{63}[/tex] → Match the denominator of the fraction
[tex]\frac{334}{63}[/tex] → Final Answer
From the results above, it can be concluded that the results of this operation 2/3 (5 × 4) ÷ 3 + 6/7 is 334/63
Learn moreInfinite Number of Solutions : https://brainly.com/question/5450548System of Equations : https://brainly.com/question/1995493System of Linear equations : https://brainly.com/question/3291576Answer detailsGrade: Middle School
Subject: Mathematics
Chapter: Order of Operations
Keywords: Linear , Equations , 1 , Variable , Line , Gradient , Point , Multiplication , Division , Exponent , PEMDAS
The answers 80kg, 88 divided by 110 then times by 100
Answers
The student is completing a two-step mathematical operation to find a percentage by dividing 88 by 110 and then multiplying the result by 100. The '80kg' mention seems to be unrelated. Without more context, it's difficult to relate this to the given references, though they involve calculations and conversions in physics.
Explanation:The student seems to be performing a two-step operation where they are first dividing 88 by 110 and then multiplying it by 100. This is a common way of finding percentages. To get the result first, divide 88 by 110 which equals 0.8. Secondly, multiply 0.8 by 100 to get 80%. The '80kg' seems irrelevant to this operation if no further context is provided, it could be a weight measurement in physics.
In terms of any associations with the provided references, it seems that the question might be related to converting some measurements or calculating kinetic energy ('K'). However, without additional information, it's hard to ascertain the exact nature of the task.
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A 5-foot ladder is leaning against a 4-foot wall. How far must the bottom of the ladder be from the base of the wall so that the top of the ladder rests on the top of the wall?
Answers
C would be the hypotenuse, which in this case is the five ft ladder.
The other known length of the leg is 4 ft.
5^2-4^2=25-16=9
9 must equal a^2 and a must equal 3
The correct answer is 3 ft
so for now the length is the unknown. therefore,
length = sqrt (5^2 - 4^2)
length = sqrt 9
length = 3 foot
3ft is the answer
Armando has a baseball card that is worth $45. The value of the card is increasing at a rate of 1.5% per year. How much will the card be worth in 15 years?