For subspace (a.1) basis for Col(A) is the first four columns of A and dimension is 4. For (a.2) basis for Row(A) is the first four rows of A and dimension is 4. A basis for (a.3) Nul(A) is the set of vectors obtained by setting one free variable to 1 and the others to 0 and dimension is 7.
A basis for a subspace is a set of vectors that are linearly independent and span the subspace. The dimension of a subspace is the number of vectors in a basis for that subspace.
(a.1) Col(A) is the subspace of R^4 spanned by the columns of A. To find a basis for Col(A), we can reduce A to its reduced row echelon form (RREF) and find the columns of A that correspond to the pivot columns in the RREF. The RREF of A is:
1 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0
The pivot columns are the first four columns, so a basis for Col(A) is the first four columns of A:
{[1, 1, 1, 1], [1, 1, -1, -1], [1, -1, 1, -1], [1, -1, -1, 1]}
The dimension of Col(A) is the number of vectors in the basis, which is 4.
(a.2) Row(A) is the subspace of R^11 spanned by the rows of A. To find a basis for Row(A), we can reduce A to its RREF and find the nonzero rows. The RREF of A is the same as above, so a basis for Row(A) is the first four rows of A:
{[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2], [0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0], [0, 0, 0, 0, 0, 0, 1, -3, 3, 0, 0]}
The dimension of Row(A) is the number of vectors in the basis, which is 4.
(a.3) Nul(A) is the subspace of R^11 consisting of all vectors x such that Ax = 0. To find a basis for Nul(A), we can reduce A to its RREF and find the solutions to the homogeneous equation Ax = 0. The RREF of A is the same as above, and the general solution to Ax = 0 is:
x1 = 0
x2 = 0
x3 = 0
x4 = 0
x5 = free
x6 = free
x7 = free
x8 = free
x9 = free
x10 = free
x11 = free
A basis for Nul(A) is the set of vectors obtained by setting one free variable to 1 and the others to 0:
{[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]}
The dimension of Nul(A) is the number of vectors in the basis, which is 7.
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Answer:
1+1= 2
Step-by-step explanation:
im a genuis
10% of 700 the simplest to do it
Answer:
70
Step-by-step explanation:
When we want to find 10% of a number, we move it's current decimal to the left one time.
700.
←
70.
10% of 700 is 70.
Enter the correct answer in the box.
Write the expression 12-2 in simplest form.
Answer:6 - 1 = 5
Step-by-step explanation: 12 / 2 = 6. 2 / 2 = 1.
I am needing some help with this
Answer:
y ≈ 27
Step-by-step explanation:
using the tangent ratio in the right triangle
tan48° = [tex]\frac{opposite}{adjacent}[/tex] = [tex]\frac{30}{y}[/tex] ( multiply both sides by y )
y × tan48° = 30 ( divide both sides by tan48° )
y = [tex]\frac{30}{tan48}[/tex] ≈ 27 ( to the nearest whole number )
Simply these expressions 2*x*3*y
The value of the expression is A = 6xy
What do you mean by an Equation?Equations are statements in mathematics that have two algebraic expressions on either side of the equals (=) sign.
It displays the similarity of the connections between the phrases on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are examples of the parts of an equation. When creating an equation, the "=" symbol and terms on both sides are necessary.
Given data ,
Let the expression be represented as A
Now , the value of A is
A = 2 ( x ) ( 3 ) ( y )
On simplifying the equation , we get
A = ( 2 ) (3 ) ( xy )
A = 6xy
Therefore , the value of A is 6xy
Hence , the expression is A = 6xy
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given the hyperbolic spiral r=1/theta at t=pi, find the slope of
the curve
A. -pi
B. pi
C. 2pi
D. -2pi
The slope of the curve at a particular point on the hyperbolic spiral, we need to take the derivative of the equation with respect to θ and evaluate it at the given value of θ. Slope of the curve found to be = 2pi. The correct answer is option C
The equation of the hyperbolic spiral is given by: r = 1/θ To express this equation in terms of x and y, we use the polar-to-rectangular coordinate transformation: x = r cos(θ) y = r sin(θ)
Substituting the equation for r, we get: x = (cos(θ))/θ y = (sin(θ))/θ Taking the derivative of y with respect to x using the chain rule, we get:
[tex](dy/dx) = (dy/dθ)/(dx/dθ) (dy/dx) = [(cos(θ)/θ^2) + (sin(θ)/θ)] / [(-sin(θ)/θ^2) + (cos(θ)/θ)] At t = π, θ = π/2.[/tex]
Therefore, the slope of the hyperbolic spiral at t = π is: (dy/dx) = 2/π The correct option to this value is C. 2pi
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pls help 20 pts and pls show work
The measure of angle IHJ is 131 degree.
What is relation between angle at the center and the angle at the circumference?The angle subtended by an arc at the center is twice the angle subtended at the circumference . More simply, the angle at the center is double the angle at the circumference.
From the figure,
3) Given that, arc ML=53 degree
∠LHM = 1/2 ×53
= 26.5
Arc MI = 89°
∠MHI=1/2 × 89°
= 44.5°
Here, ∠IHJ+∠MHI+∠LHM+∠LHK+∠KHJ=360°
∠IHJ+44.5°+26.5°+88°+70°=360°
∠IHJ=360°-229°
∠IHJ=131°
4) Arc WV= 1/2 ×55° = 27.5°
Here, Arc WT = Arc TU + Arc UV + Arc WV
= 50°+100°+27.5°
= 177.5°
5) Here, HG= 1/2 ×40 =20°
Arc FH= Arc FG+ Arc HG°
= 60°+20°
= 80°
Therefore, the measure of angle IHJ is 131 degree.
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log_4(xy^5z^4)
Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.)
The expanded form of the expression as a sum constant multiple of logarithms is:
log₄(xy⁵z⁴) = log₄(x) + 5log₄(y) + 4log₄(z)
What are the properties of logarithms?There are four basic properties of logarithms:
logₐ(xy) = logₐx + logₐy.
logₐ(x/y) = logₐx - logₐy.
logₐ(xⁿ) = n logₐx.
logₐx = logₓa / logₓb.
According to the problem, we will use some of the basic logarithmic properties,
Given expression,
log₄(xy⁵z⁴)
We can use the properties of logarithms to expand the expression:
log₄(xy⁵z⁴) = log₄(x) + log₄(y⁵) + log₄(z⁴)
Using the properties of logarithms, we can separate the terms inside the logarithm as separate logarithms, where the multiplication of variables is represented as a sum of logarithms.
So, we have:
log₄(xy⁵z⁴) = log₄(x) + 5log₄(y) + 4log₄(z)
Therefore, the expanded form of the expression as a sum constant multiple of logarithms is:
log₄(xy⁵z⁴) = log₄(x) + 5log₄(y) + 4log₄(z)
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Im stuck on this problem. 25 points for the help!
Answer:
Step-by-step explanation:
First we can determine that this is a special 45-45-90 triangle, since the two legs are congruent, meaning the two unknown angles are congruent as well:
180 = 90 + 2(theta)
theta = 45
Knowing this, we can remember that the hypotenuse of this special triangle is equal to the leg*sqrt(2).
This means that 1 = x(sqrt(2)).
[tex]x=\frac{1}{\sqrt{2} }[/tex]
Hope this helps!
What is one hundredth less than 5.31?
Answer:
5.30
Step-by-step explanation:
One hundredth is 0.01. 5.31-0.01=5.30.
sorry for answering late!
(1 point) Determine whether the following matrices are in echelon form, reduced echelon form or not in echelon form.
a. Reduced echelon form [100 010 000 -10-100]
b. Not in echelon form [ -8000 -4200 -8110 -9101 -8130]
c. Reduce echelon form [11 10 -4 -10]
d. Echelon form [ 100 001 000 -502]
The given matrices are checked and their status is mentioned in the following list below :a. Reduced echelon form [100 010 000 -10-100] b. Not in echelon form [ -8000 -4200 -8110 -9101 -8130] c. Reduce echelon form [11 10 -4 -10] d. Echelon form [ 100 001 000 -502] Echelon form and reduced echelon form of matrices
The Matrix A is in the reduced echelon form if and only if the following conditions are satisfied:All rows having 0s are at the bottom of the matrix.There must be no 0 rows in the matrix.In each row of the matrix containing the leading nonzero element, all entries above and below the leading entry are 0.
The leading coefficient of a nonzero row is always strictly to the right of the leading coefficient of the row above it.In each column containing a leading coefficient, all other entries are zero.
For a matrix to be in echelon form, it must fulfill the following conditions : All rows containing all zero elements are located at the bottom of the matrix. All leading coefficients in any nonzero row are always strictly to the right of the leading coefficient in the row above it.
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PLEASE HELP ASAP
A scatter plot is shown on the coordinate plane.
scatter plot with points plotted at 1 comma 2, 1 comma 3, 3 comma 2, 3 comma 3, 4 comma 2, 4 comma 5, 5 comma 6, 6 comma 6, 7 comma 8, 8 comma 7, 9 comma 10, and 10 comma 9
Which of the following graphs shows a line on the scatter plot that fits the data?
scatter plot with points plotted at 1 comma 2, 1 comma 3, 3 comma 2, 3 comma 3, 4 comma 2, 4 comma 5, 5 comma 6, 6 comma 6, 7 comma 8, 8 comma 7, 9 comma 10, and 10 comma 9, with a line drawn through the points 6 comma 6 and 7 comma 7
scatter plot with points plotted at 1 comma 2, 1 comma 3, 3 comma 2, 3 comma 3, 4 comma 2, 4 comma 5, 5 comma 6, 6 comma 6, 7 comma 8, 8 comma 7, 9 comma 10, and 10 comma 9, with a line drawn through the points 4 comma 5 and 5 comma 6
scatter plot with points plotted at 1 comma 2, 1 comma 3, 3 comma 2, 3 comma 3, 4 comma 2, 4 comma 5, 5 comma 6, 6 comma 6, 7 comma 8, 8 comma 7, 9 comma 10, and 10 comma 9, with a line drawn through the points 3 comma 3 and 4 comma 4
scatter plot with points plotted at 1 comma 2, 1 comma 3, 3 comma 2, 3 comma 3, 4 comma 2, 4 comma 5, 5 comma 6, 6 comma 6, 7 comma 8, 8 comma 7, 9 comma 10, and 10 comma 9, with a line drawn through the points 1 comma 2 and 7 comma 8
Answer:
scatter plot with points plotted at 1 comma 2, 1 comma 3, 3 comma 2, 3 comma 3, 4 comma 2, 4 comma 5, 5 comma 6, 6 comma 6, 7 comma 8, 8 comma 7, 9 comma 10, and 10 comma 9, with a line drawn through the points 4 comma 5 and 5 comma 6.
Step-by-step explanation:
The scatter plot shows a collection of points on a coordinate plane. To determine which of the given graphs shows a line that fits the data, we need to analyze the points and their distribution.
From the given points, we can observe that there is no clear linear relationship between the x and y variables. However, we can see that there is a cluster of points between (4,2) and (7,8), and the rest of the points are scattered around this cluster. Therefore, any line that fits the data should pass through this cluster of points.
Now, let's analyze each option:
The first option shows a line passing through (6,6) and (7,7). While these two points are part of the cluster, the line does not seem to fit the rest of the data very well.
The second option shows a line passing through (4,5) and (5,6). These points are also part of the cluster, and the line seems to fit the data reasonably well.
The third option shows a line passing through (3,3) and (4,4). These points are near the edge of the cluster, but the line seems to fit the data reasonably well.
The fourth option shows a line passing through (1,2) and (7,8). While these two points are part of the cluster, the line does not seem to fit the rest of the data very well.
Based on the analysis above, the second and third options seem to be the best fits for the data. However, the second option seems to be a slightly better fit than the third option since the line passes through two points that are closer to the center of the cluster. Therefore, the answer is:
scatter plot with points plotted at 1 comma 2, 1 comma 3, 3 comma 2, 3 comma 3, 4 comma 2, 4 comma 5, 5 comma 6, 6 comma 6, 7 comma 8, 8 comma 7, 9 comma 10, and 10 comma 9, with a line drawn through the points 4 comma 5 and 5 comma 6.
Answer: B
scatter plot with points plotted at 1 comma 2, 1 comma 3, 3 comma 2, 3 comma 3, 4 comma 2, 4 comma 5, 5 comma 6, 6 comma 6, 7 comma 8, 8 comma 7, 9 comma 10, and 10 comma 9, with a line drawn through the points 4 comma 5 and 5 comma 6
Step-by-step explanation:
Rectangles R and S are similar. If the area of rectangle R is 187, what is the area of rectangle S?
The area of rectangle S is 84.15 square unit.
What do we mean by area?Area is the region enclosed by the shape of an object. The space covered by a figure or any two-dimensional geometric shape in the plane is the area of the shape, or
Area is a measure of the surface area of a shape. To find the area of a rectangle or square, you need to multiply the length and width of the rectangle or square.
Solution according to the given information in the question:
Area of rectangle R = 187
Ratio of rectangle R = 17
Ratio of rectangle S = 7.65
Area of rectangle S = (7.65/17)×187
= 84.15 square unit
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What is the end behavior of y as x goes to negative infinity in
the equation y = -3x^5 + 4x^2 – 7?
The end behavior of y as x goes to negative infinity in the equation y = -3x^5 + 4x^2 – 7 can be determined by looking at the leading term of the polynomial, which is -3x^5. As x goes to negative infinity, this term will dominate the other terms and determine the end behavior of the function.
Since the degree of the leading term is 5, which is an odd number, and the coefficient is negative, the end behavior of the function will be that y goes to positive infinity as x goes to negative infinity. This can be written in notation as:
lim(x→-∞) y = +∞
In conclusion, the end behavior of y as x goes to negative infinity in the equation y = -3x^5 + 4x^2 – 7 is that y goes to positive infinity.
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The pressure of a gas varies jointly as the amount of the gas (measured in moles) and the temperature and inversely as the volume of the gas. If the pressure is 1170 kpa (kiloPascals) when the number of moles is 6, the temperature is 260 kelvin�, and the volume is�720 cc, find the pressure when the number of moles is 3, the temperature is�280 k, and the volume is 180 cc
The formula for the pressure of a gas is given by the Ideal Gas Law, which states that P = nRT/V, where P is the pressure, n is the amount of gas in moles, R is the ideal gas constant, T is the temperature in kelvin, and V is the volume in cubic centimeters (cc).
Given the initial conditions, we can find the value of the ideal gas constant R:
1170 kpa = (6 moles)(R)(260 K) / (720 cc)
R = (1170 kpa)(720 cc) / (6 moles)(260 K)
R = 8.27 kpa·cc/mol·K
Now, we can use this value of R to find the pressure under the new conditions:
P = (3 moles)(8.27 kpa·cc/mol·K)(280 K) / (180 cc)
P = 3898.6 kpa·cc / 180 cc
P = 2166 kpa
Therefore, the pressure of the gas when the number of moles is 3, the temperature is 280 K, and the volume is 180 cc is 2166 kpa.
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(a) [Marks=5] Construct a one replicate of a 24 factorial design in factors A, B, C and D in blocks of size 4 by confounding the interactions ABC and ACD with the blocks. Show your steps clearly. Listing the contents of two blocks, one of which contains the treatment combination (1) and the other contains treatment combination a, will suffice the purpose.
(b) [Marks=1] What other interactions, if any, are confounded with the blocks.
Other interactions that are confounded with the blocks are BCD and BCD.
(a) Constructing a one replicate of a 24 factorial design in factors A, B, C, and D in blocks of size 4 by confounding the interactions ABC and ACD with the blocks:
1. Number the blocks from 1 to 4.
2. Assign the factor levels to each block, so that the interactions ABC and ACD are confounded. One possible way to do this is as follows:
3. List the contents of two blocks, one of which contains the treatment combination (1) and the other contains treatment combination a.
(b) Other interactions that are confounded with the blocks are BCD and BCD.
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I'LL GIVE YOU BRANLIST
Answer:
1. plot the point (6,5) and label it Mr. Snuffles
2. 26 treats were eaten in total (21 not counting Mr.Snuffles)
Step-by-step explanation for plotting the point:
On the bottom line (miles walked) go to the number six, then go up five until it is on the five line for the side line (number of treats) plot the point here and then label the point Mr.Snuffles.
plot the point (6,5)
21 treats were eaten in total
Step-by-step explanation:
On the X axis, go to the number six, then go up five until it is on the five line for the side line (number of treats) plot the point here and then label the point Mr. Snuffles.
Si tengo un cubo de 3600 m³ cuantos kilo litros caben?
If you have a 3600 m³ cube, then you can fit 3600 kiloliters in the cube.
A kiloliter (kL) is defined as a unit of volume which is used to measure liquids. It is equal to 1,000 liters, and it is used to measure large volumes of water, oil, and other liquids.
We know that 1 cubic meter is equal to 1 kiloliter,
We have to convert 3600 meter cube to Kiloliter,
So , we multiply by 3600,
On multiplying,
We get,
⇒ 3600 meter cube = 3600 × 1 Kiloliter,
⇒ 3600 meter cube = 3600 Kiloliter,
Therefore, 3600 Kiloliter can fill in the cube having the volume as 3600 meter cube.
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Jack swam 3. 4 miles in the same time that Christine swam 4. 1 miles. How many more miles did Christine swim?
Christine swam 0.7 miles more than Jack.
The solution can be simply found out by subtracting the distance swam by Jack by the distance swam by Christine.
Distance swam more by Christine= 4.1 - 3.4
Distance swam more by Christine= 0.7
The mile is a customary unit of measurement in the United States and the United Kingdom that is based on the older English unit of length equal to 5,280 English feet, or 1,760 yards. It is often referred to as the international mile or statute mile to distinguish it from other miles. One mile can be covered in under one minute. Nonetheless, there are varying speed limits on the roads. You should include in their average speed of 25–60 mph when calculating how long it will take you to reach your destination.
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Based on the following figure, determine which value below is correct.
The correct measure is given as follows:
x = 6.
What are vertical angles?Vertical angles are angles that are opposite by the same vertex on crossing segments, hence they share a common vertex, thus being congruent, meaning that they end up having the same angle measure.
Over vertex C, the two angles are congruent, hence the value of x is obtained as follows:
12x - 9 = 6x + 27
6x = 36
x = 6.
(as the vertical angles are congruent, we can just equal the measures of the two angles and then solve the expression for the value of x).
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homeworkld 617158840&questionid=18 flushed=false&cid=68150978.centerwin MATH 1762 Precalculus section 61 - Spring 2022 mohamed musse 03/06/22 Homework: Homework 6 - Part II Question 3, 6.4.9 HW Score:
Find the domain of the logarithmic function y = log (5x+6) analytically. You may wish to check your answer graphically The domain is (____, [infinity]) (Simplify your answer. Type an integer or fraction)
For the logarithmic function y = log (5x+6), the domain is (-6/5, [infinity]).
The domain of a logarithmic function is the set of all values for which the function is defined. In the case of y = log(5x+6), the function is only defined for values of x that make the expression inside the logarithm, 5x+6, greater than zero. This is because the logarithm of a negative number or zero is not defined.
To find the domain analytically, we need to solve the inequality 5x+6 > 0 for x:
5x+6 > 0
5x > -6
x > -6/5
This means that the domain of the function is all values of x greater than -6/5. In interval notation, this can be written as (-6/5, infinity).
So the domain of the logarithmic function y = log(5x+6) is (-6/5, infinity).
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Work out the cincumference of a cylinder with a 6 cm radius and a 4 cm height
In the following question, among the conditions given, the circumference of a cylinder is "20π cm, or approximately 62.83 cm" with a 6 cm radius and a 4 cm height.
The formula for the circumference of a cylinder is:
Circumference = 2πr + 2πh
where r is the radius of the circular base and h is the height of the cylinder.
In this case, the radius is 6 cm and the height is 4 cm, so we can plug those values into the formula:
Circumference = 2π(6) + 2π(4)
Circumference = 12π + 8π
Circumference = 20π
So the circumference of the cylinder is 20π cm, or approximately 62.83 cm (if we use a calculator to approximate the value of π to two decimal places).
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Use the function h(x)=−2x+8 to answer the
following questions
Evaluate h(−9):
h(−9)= ____
For what values(s) of x does h(x)=−10?
x= ___
The value of h(-9)= 26 using the function h(x)=−2x+8
And, the value of x for which h(x) = -10 is x = 9.
To evaluate h(-9), we simply plug in -9 for x in the function h(x) = -2x + 8:
h(-9) = -2(-9) + 8
h(-9) = 18 + 8
h(-9) = 26
So the value of h(-9) is 26.
To find the value(s) of x for which h(x) = -10, we can set the function equal to -10 and solve for x:
-2x + 8 = -10
-2x = -18
x = 9
So the value of x for which h(x) = -10 is x = 9.
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System of Equations
y = (x-2)² + 35
y = -2x + 15
been trying to figure this one out for hours, please help
The solution of the system of equations is ([2+9.48i]/2, 17+9.48i).
What is a linear system of equations?A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously. The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.
The given system of equations are y=(x-2)²+35 ----(i) and y=-2x+15 ----(ii).
Here, equation (i) = (ii)
(x-2)²+35= -2x+15
x²-4x+4+35= -2x+15
x²-4x+39= -2x+15
x²-4x+39+2x-15=0
x²-2x+24=0
x²-2x+24=0
By using quadratic formula, we get
x = [-b ± √(b² - 4ac)]/2a
x=[2±√((-2)² - 4×1×24)]/2×1
x=[2±√(-90)]/2
x=[2±9.48i]/2
Here, x=[2+9.48i]/2 and x=[2-9.48i]/2
So, y=2+9.48i+15=17+9.48i
Therefore, the solution of the system of equations is ([2+9.48i]/2, 17+9.48i).
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A rectangular swimming pool in a small recreation centre can hold 60 000 L of water. A new, larger recreation centre is being built with a pool that is a scaled-up (enlarged) version of the smaller centre's pool. If the new pool can hold 3 039 180 L of water, what was the scale factor from the original pool's dimensions (length, width, and height) to the new pool's dimensions?
The scale factor from the original pool's dimensions to the new pool's dimensions is approximately 3.682.
The volume of a rectangular prism (such as a swimming pool) is given by the formula V = lwh, where l is the length, w is the width, and h is the height.
Let x be the scale factor from the original pool's dimensions to the new pool's dimensions. Then, the dimensions of the new pool can be expressed as lx, wx, and hx, where l, w, and h are the dimensions of the original pool.
We know that the original pool can hold 60,000 L of water, so we can set up the equation:
60,000 = lwh
Substituting in lx for l, wx for w, and hx for h, we get:
60,000 =[tex](lx)(wx)(hx)[/tex]
We also know that the new pool can hold 3,039,180 L of water, so we can set up another equation:
3,039,180 = [tex](lx)(wx)(hx)[/tex]
The result of dividing the second equation by the first equation is:
3,039,180/60,000 = [tex](lx)(wx)(hx)/(lx)(wx)(hx)[/tex]
Simplifying, we get:
50.653 =[tex]x^3[/tex]
The result of taking the cube root of both sides is:
x ≈ 3.682
Therefore, the scale factor from the original pool's dimensions to the new pool's dimensions is approximately 3.682.
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Which of the following sets represents the solution of the equation below?
-3/2x² = x+1
The solution set is: [tex]$\left{\frac{-1+\sqrt{5}}{3}, \frac{-1-\sqrt{5}}{3}\right}$[/tex]
Option (D) is correct.
What is a quadratic equation?
A quadratic equation is a type of equation in algebra that can be written in the form of ax^2 + bx + c = 0, where x is the unknown variable, and a, b, and c are constants with a not equal to zero
The equation is:
[tex]$-\frac{3}{2}x^2 = x + 1$[/tex]
We can rewrite this equation as:
[tex]$-\frac{3}{2}x^2 - x - 1 = 0$[/tex]
To solve for x, we can use the quadratic formula:
[tex]$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$[/tex]
Where a = -3/2, b = -1, and c = -1. Substituting these values into the quadratic formula, we get:
[tex]$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(-\frac{3}{2})(-1)}}{2(-\frac{3}{2})}$[/tex]
Simplifying this expression, we get:
[tex]$x = \frac{1 \pm \sqrt{5}}{3}$[/tex]
Therefore, the solution set is:
[tex]$\left{\frac{-1+\sqrt{5}}{3}, \frac{-1-\sqrt{5}}{3}\right}$[/tex]
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Explain how to use equivalent fractions to compare the fractions 3/4 7/12
To compare fractions using equivalent fractions, we need to find a common denominator, convert each fraction to an equivalent fraction with that denominator, and then compare the numerators directly.
To compare fractions 3/4 and 7/12 using equivalent fractions, we need to find a common denominator for both fractions. A common denominator is a number that is divisible by all the denominators of the fractions being compared. In this case, the least common denominator (LCD) for 4 and 12 is 12.
To convert 3/4 to an equivalent fraction with a denominator of 12, we need to multiply both the numerator and denominator by 3. This gives us the equivalent fraction of 9/12.
To convert 7/12 to an equivalent fraction with a denominator of 12, we need to multiply both the numerator and denominator by 1. This gives us the equivalent fraction of 7/12.
Now that both fractions have the same denominator, we can compare them directly. We can see that 9/12 is greater than 7/12 since 9 is greater than 7. Therefore, 3/4 is greater than 7/12.
This method allows us to compare fractions with different denominators and make accurate comparisons based on the relative size of the numerators.
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Find
f ∘ g
and
g ∘ f.
f(x) =
3 x − 8
, g(x) = x3 + 1
(a)
f ∘ g
(b)
g ∘ f
Find the domain of each function and each composite function. (Enter your answers using interval notation.)
domain of f
domain of g
domain of f ∘ g
domain of g ∘ f
The domains of each function and each composite function are as follows:
domain of f: (-∞, ∞)
domain of g: (-∞, ∞)
domain of f ∘ g: (-∞, ∞)
domain of g ∘ f: (-∞, ∞)
To find the composite functions f ∘ g and g ∘ f, we need to substitute the function g(x) into f(x) for f ∘ g, and the function f(x) into g(x) for g ∘ f.
(a) f ∘ g = f(g(x)) = f(x3 + 1) = 3(x3 + 1) − 8 = 3x3 + 3 − 8 = 3x3 − 5
(b) g ∘ f = g(f(x)) = g(3x − 8) = (3x − 8)3 + 1 = 27x3 − 72x2 + 64x − 511
The domain of a function is the set of all values of x for which the function is defined.
The domain of f is all real numbers, since there are no restrictions on the values of x. So the domain of f is (-∞, ∞).
The domain of g is also all real numbers, since there are no restrictions on the values of x. So the domain of g is (-∞, ∞).
The domain of f ∘ g is the same as the domain of g, since the values of x are first substituted into g(x) before being substituted into f(x). So the domain of f ∘ g is (-∞, ∞).
The domain of g ∘ f is the same as the domain of f, since the values of x are first substituted into f(x) before being substituted into g(x). So the domain of g ∘ f is (-∞, ∞).
Therefore, the domains of each function and each composite function are as follows:
domain of f: (-∞, ∞)
domain of g: (-∞, ∞)
domain of f ∘ g: (-∞, ∞)
domain of g ∘ f: (-∞, ∞)
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following relation. - 6x2 - 5y = 4x + 3y Determine the implied domain of the function found in the first step. Express your answer in interval notation. Points
The implied domain of the function y = (-6x^2 - 4x)/8 is found to be (-∞, ∞).
The first step in determining the implied domain of the function is to isolate the dependent variable, in this case y, on one side of the equation.
We can do this by adding 5y to both sides of the equation and subtracting 4x from both sides of the equation:
-6x^2 - 4x = 8y
Next, we can divide both sides of the equation by 8 to get y by itself:
(-6x^2 - 4x)/8 = y
Now we have the function in terms of y:
y = (-6x^2 - 4x)/8
The implied domain of the function is the set of all x-values for which the function is defined.
In this case, there are no restrictions on the values of x that can be used in the function, so the implied domain is all real numbers.
Therefore, the implied domain of the function y = (-6x^2 - 4x)/8 is (-∞, ∞).
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I need help with number 1 the tan= 4/3, lies in quadrant ||| Find sin20
It’s due tonight at 11:59 and I’m struggling
Step-by-step explanation:
Step 1: Simplify the identity we need to find
[tex] \sin(2x) = 2 \sin(x) \cos(x) [/tex]
So we need to find sin and cos
Here, we are given tan(x)
A simple way to find sin (x) and cos(x) when given tan(x) is to use the definition of the trig functions of acute angles.
[tex] \tan( \alpha ) = \frac{y}{x} [/tex]
[tex] \cos( \alpha ) = \frac{x}{r} [/tex]
[tex] \sin( \alpha ) = \frac{y}{r} [/tex]
where r is
[tex]r = \sqrt{ {x}^{2} + {y}^{2} } [/tex]
Here, y is 4 and x is 3.
So
[tex]r = \sqrt{ {3}^{2} + {4}^{2} } = 5[/tex]
Since both cosine and sine are in quadrant 3, they are both negative.
Know we can plug in the knowns, since we know x,y, and r.
[tex] \cos( \alpha ) = - \frac{3}{5} [/tex]
[tex] \sin( \alpha ) = - \frac{4}{5} [/tex]
Now plug in the knowns for sin 2a
[tex] \sin(2 \alpha ) = 2( - \frac{3}{5} )( - \frac{4}{5} ) = \frac{24}{25} [/tex]
Help, please. I'm so confused and every time I work on it I get it wrong and other people aren't helping.
The value of the variable 'x' in the isosceles right-angle triangle will be 2√2 units.
What is a Pythagoras theorem?The Pythagoras theorem states that the sum of two squares equals the squared of the longest side.
The Pythagoras theorem formula is given as,
H² = P² + B²
The length of the hypotenuse is 4. And the triangle is an isosceles right triangle.
In the isosceles right triangle, the perpendicular and base of the triangle will be the same. Then the value of the variable 'x' is given as,
4² = x² + x²
2x² = 16
x² = 8
x = 2√2
The value of the variable 'x' in the isosceles right-angle triangle will be 2√2 units.
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