To calculate the kernel, locate the vector subspace in which the homogeneous implicit equations result from the linear transformation formula's component parts being equal to zero.
What is linear transformation?A function from one vector space to another that respects the underlying (linear) structure of each vector space is called a linear transformation.
A linear operator, or map, is another name for a linear transformation.
The zero transformation and identity transformation are two significant illustrations of linear transformations.
An illustration of a linear transformation is the zero transformation, which is denoted by T(x)=(0) for every x.
Also linear is the identity transformation denoted by T(x)=(x).
Find the vector subspace in which the implicit equations are the homogeneous equations produced when the components of the linear transformation formula are equal to zero in order to compute the kernel.
This is the same as finding the null space of the linear transformation matrix.
Therefore, to calculate the kernel, locate the vector subspace in which the homogeneous implicit equations result from the linear transformation formula's component parts being equal to zero.
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problem 3: compute the general solution for: d 2 y dt2 − 2 dy dt − 15y = e 4t
The general solution for given equation is y = c₁e^5t + c₂e^-3t - (1/7)e^4t.
A differential equation is an equation that contains at least one derivative of an unknown function, either an ordinary derivative or a partial derivative. Suppose the rate of change of a function y with respect to x is inversely proportional to y, we express it as dy/dx = k/y.
In calculus, a differential equation is an equation that involves the derivative (derivatives) of the dependent variable with respect to the independent variable (variables). The derivative represents nothing but a rate of change, and the differential equation helps us present a relationship between the changing quantity with respect to the change in another quantity. y=f(x) be a function where y is a dependent variable, f is an unknown function, x is an independent variable.
We have to find the general solution for d²y/dt² - 2 dy/dt - 15y = e^4t
Solving the above ordinary differential equation, we get:
y = c₁e^5t + c₂e^-3t - (1/7)e^4t
Thus, the general solution for given equation is y = c₁e^5t + c₂e^-3t - (1/7)e^4t
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A random sample of 9th-grade students was asked if they prefer taking notes on a computer or using a pencil. Of the 180 students who were surveyed, 75 said they preferred using the computer. The resulting 99% confidence interval of the proportion of students who prefer taking notes on the computer was (0. 322, 0. 511). The school newspaper ran a story saying that less than half of the body prefers taking notes on a computer. Based on the interval, is the newspaper justified in this statement?
The newspaper's assertion is unjustified. It can be shown that the interval does not contain 0.5 based on the 99% confidence interval of (0.322, 0.511). (which is half).
This indicates that the population parameter (mean, percentage, and standard deviation) is between a and b with x% confidence.
This indicates that at least half of students prefer using a computer to take their notes. Since the interval's lower limit (0.322) is higher than 0.5, there is a 99% likelihood that the actual percentage of students who prefer taking notes on computers is higher than 0.5.
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8. (6 pts) At a local garden shop, the price of plants includes sales tax. The cost of 4 large plants and 8
medium plants is $40. The cost of 5 large plants and 2 medium plants is $28.
b) Could the cost of one large plant be $5.50 and the cost of one medium plant be $2.25? Justify your
answer.
c)
Determine algebraically both the cost of a large plant and the cost of a medium plant
By solving a system of equations we will see that a large plant costs $4.50 and a medium one $2.75
How to find the costs of the plants?Let's define the variables:
x = cost of a large plant.
y = cost of a medium plant.
We know that " The cost of 4 large plants and 8 medium plants is $40"
4*x + 8*y = 40
And "The cost of 5 large plants and 2 medium plants is $28."
5*x + 2*y = 28
So we have a system of equations:
4*x + 8*y = 40
5*x + 2*y = 28
We can isolate x on the first equation to get:
4x + 8y = 40
4x = 40 - 8y
x = (40 - 8y)/4
x = 10 - 2y
Replacing that in the other equation we will get:
5*(10 - 2y) + 2*y = 28
50 - 10y + 2y = 28
50 - 28 = 8y
22 = 8y
22/8 = y
2.75 = y
And the value of x is:
x = 10 - 2*2.75 = 4.5
Then the cost of a lare plant is $4.5 and the cost of a medium plant is $2.75.
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Which function has a percent rate of decrease equal to 5%?
A. F(x) = 3(0. 5)*
B. F(x) = 3(1. 5)
C. F(x) = 3(0. 05)
D. F (x) = 3(0. 95)
E. F(x) = 3(1. 05)
Among the given, the function that has a percent rate of decrease equal to 5% is: F(x) = 3(0.95)
Hence, option (D) is the correct choice.
For (A):
We have,
F(x) = 3(0.5), Here percentage rate decrease = 50%
For (B):
We have,
F(x) = 3(1.5), Here there is percentage rate increase which will be = 50%
For (C):
We have,
F(x) = 3(0.05), Here percentage rate decrease = 95%
For (D)
F(x) = 3(0.95)
Here, percent rate of decrease equal to 5%
This function represents the relationship between input x and the output y where y = 3(0.95^x).
The number 0.95 can be interpreted as 95% of the original value, which means a 5% decrease.
For (A):
We have,
F(x) = 3(1.05), Here there is percentage rate increase which is = 5%
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The CEO of the Wild Widget Company has decided to invest $360, 000 in his Michigan facgory. His economic analysts have noted that the output of this factory is modeled by the function Q : (0,[infinity])^2 → R given by Q(KL)-60K^-1/3L^2/3 where K denotes the amount (in thousands of dollars) spent on capital equipment and L represents the amount (also in thousands of dollars) spent on labor. (a) How should the CEO allocate the $360, 000 between labor and equipment? (b) Show that
aQ/aK=aQ/aL
at the point (K, L) found in part (a)
xplanation:
a b c your way out the picture
The CEO should allocate $144,000 to labor and $216,000 to capital equipment.
How should the CEO allocate the $360, 000 between labor and equipment?Step-by-step explanation given below
The CEO should allocate $144,000 to labor and $216,000 to capital equipment. This allocation maximizes the output of the factory, which can be found by taking the partial derivatives of the function Q with respect to K and L and setting them equal to 0.
Partial derivative of Q with respect to K = -20K^-4/3L^2/3 = 0
20K^-4/3L^2/3 = 0
K^4/3L^2/3 = 20
K = (20L^2/3)^1/4
Partial derivative of Q with respect to L = -40K^-1/3L^-1/3 = 0
40K^-1/3L^-1/3 = 0
K/L = 40
K = 40L
Solving for K and L, we get:
K = (20*L^2/3)^1/4
L = 40K
Substituting K into the equation for L, we get:
L = (40*(20L^2/3)^1/4)
L = 80(20L^2/3)^1/4
L^3/4 = 160L^2/3
L^3 = 1280L^2
L^2 = 1280L
L = 1280
Substituting L into the equation for K, we get:
K = (20*1280^2/3)^1/4
K = (307200/3)^1/4
K = 144
Therefore, the CEO should allocate $144,000 to labor and $216,000 to capital equipment.
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The equation of Line D is y = 3/8x +3. The equation of the line E is y = 3/8x + 8/3, Are line D and Line E perpendicular,parallel or neither.
The slope of line D and line E will be congruent. Then the lines are parallel to each other.
What is a linear equation?A connection between a number of variables results in a linear model when a graph is displayed. The variable will have a degree of one.
The linear equation is given as,
y = mx + c
Where m is the slope of the line and c is the y-intercept of the line.
If the slopes are different then the lines are intersecting.If the slopes are equal but have different intercepts then the lines are parallel.If the slopes and intercepts are equal then the lines are coincident.If the product of the slope is negative then the lines are perpendicular.The equations are given below.
Line D: y = (3/8)x +3
Line E: y = (3/8)x + 8/3
The slope of line D and line E will be congruent. Then the lines are parallel to each other.
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PLS HELP WILL GIVE ALOT OF POINTS DOMAIN AND RANGE
The range of the function lies between -∞<y<∞ and the domain of the function is given by -1<x<3
What are functions?Function, in mathematics, is an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).
Given here: The graph of a function
We know The domain of a function is the set of values that we are allowed to plug into our function. This set is the x values in a function such as f(x). The range of a function is the set of values that the function assumes. This set is the values that the function shoots out after we plug an x value in.
It is clear from the graph that y ∈ (-∞,∞) and x∈ (-1,3)
we know a function is given by y=f(x)
where y is the range values and x is the domain.
Hence, The range of the function lies between -∞<y<∞ and the domain of the function is given by -1<x<3
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-5x+15
5x-25
4a+13
5x-25
The value of the equations if they are zero each are: 3, 5, -13/4 and 5
What is an equation?By definition, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =
The given equations are
-5x+15=0
-5x = -15
Making x the subject of the relation we have
x=3
2) 5x-25 = 0
Collecting like terms
5x=25
Making x the subject
x=5
3) 4a+13=0
4a = -13
Making a the subject of the relation
a=-13/4
4) 5x-25=0
Collecting like terms
5x=25
Making x the subject of the relation
x=5
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The formula for the perimeter of a rectangle is. Part a. Rewrite the formula for the perimeter of a rectangle in terms of the width. In your final answer, include all of your work. Part b. In two or more complete sentences, describe the process you followed while isolating the variable in the equation.
The isolation of variable in the equation will result in the formula - Breadth = (Perimeter - 2length)/2.
As per the known information, the formula of the perimeter of rectangle is -
Perimeter = 2 (length + breadth)
Expanding the formula we get -
Perimeter = 2length + 2breadth
Rewriting in terms of breadth -
2Breadth = Perimeter - 2length
Breadth = (Perimeter - 2length)/2
The process to isolate the variable will require expanding and rewriting the equation with breadth on Left Hand Side of the equation. Now, move the constant on Right Hand Side and you will get the formula.
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Mainsail
Headsail
Welcome to the land down under! Spend the day sailing
in Sydney Harbor! Sailboats are made up of two sails,
the mainsail and the headsail. Typically, these sails are
the shape of triangles and these two triangles are
similar. Considering the sailboat pictured below,
determine the value of x and y and the unknown side
lengths.
6+A
X-2
30 ft
5y-3
26 ft
By using the properties of similar triangles, the values of x and y are respectively, 15 & 7
What is the similarity of triangles?Triangles with the same shape but different sizes are said to be similar triangles. Squares with any side length and all equilateral triangles are examples of related objects. In other words, if two triangles are similar, their corresponding sides are proportionately equal and their corresponding angles are congruent. Triangle resemblance is shown here by the symbol "≈"
Given that,
Side lengths of triangle 1, y + 9, x, x -2
Side lengths of triangle 2, 30, 26, 5y - 3
Both the triangles are similar,
So, we can write
(x - 2)/x = 26/30
After solving it, x = 15
Similarly,
(y + 9)/x = (5y - 3)/30
y = 7
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Jayden is going to invest $97,000 and leave it in an account for 16 years. Assuming the interest is compounded annually, what interest rate, to the nearest hundredth of a percent, would be required in order for Jayden to end up with $143,000?
An interest rate of 2.5 % is required to obtained a current capital of $ 143,000 after a period of 16 years.
How to find the interest rate of a deposit
In this problem we find that Jayden deposits an amount of money for a period of 16 years and such account is under composite interest, that is, that the account increases its capital continously in time. Compound interest model is shown:
C' = C · (1 + r / 100)ˣ
Where:
C - Initial capital, in monetary units.C' - Current capital, in monetary units. r - Interest rate, in percentage.x - Number of periods, in years.If we know that C = 97,000, C' = 143,000 and x = 16, then the interest rate is:
143,000 = 97,000 · (1 + r / 100)¹⁶
143 / 97 = (1 + r / 100)¹⁶
1.025 = 1 + r / 100
0.025 = r / 100
r = 2.5
The needed interest rate needed for a capital of $ 143,000 after a period of 16 years is equal to 2.5 %.
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I need help ASAP. I can’t figure out the answer can someone give it or explain the answer?
1.5 cm is the data range. The number sequence "4, 6, 9, 3, 7" is an example where the lowest value is 3 and the highest is 9. The range is therefore 9 3 = 6. Just like that!
What is meant by data range?In statistics, the range of your data is the range between the lowest and greatest values of the distribution. It acts as a typical sign of unpredictable behavior.Measurements of variability provide descriptive statistics for your data set's summary, together with measures of central tendency. In a list or collection of numbers, the range is the difference between the lowest and highest integers. Place all the numbers in order before attempting to determine the range. The lowest number is then subtracted (subtracted) from the greatest number. The range is the range of values, i.e., the range between the lowest and highest values. The lowest value is 3 and the highest is 9 in the example number sequence "4, 6, 9, 3, 7." As a result, the range is 9 3 = 6. Simple as that!Q3 - Q1 is the interquartile range (IQR).
The third quartile (Q3) of the box plot is the data value that exactly matches the end of the box's edge (7 cm).
The data value exactly at the box's outside edge (5.5 cm) is the first quartile (Q1).
Range between quartiles (IQR): 7 - 5.5 = 1.5 cm
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Use the gcf and the distributive property to find the sum of 15+36
Answer: br uh is this even a high school question? anyway, look at the image that i sent you can see the answer there.
better give me brainliest
Step-by-step explanation:
Which statement best explains if the graph correctly represents the proportional relationship y = −2x? (4 points) A coordinate plane is shown. Points are graphed at 2 comma 4 and negative 1 comma 2. The points are joined by a line. a No, the points shown would not be part of y = −2x b No, proportions cannot be represented on a graph c Yes, all proportions can be shown on a graph of this line d Yes, the points shown on the line would be part of y = −2x
A statement which best explains if the graph correctly represents the proportional relationship y = −2x include the following: D. Yes, the points shown on the line would be part of y = −2x.
What is a proportional relationship?In Mathematics, a proportional relationship can be defined as a type of relationship that generates equivalent ratios, a straight line, and it can be modeled or represented by the following mathematical expression:
y = kx
Where:
k is the constant of proportionality.y and x represent the variables in a proportional relationship.Generally speaking, the graph of any proportional relationship such as the linear equation (y = -2x) is characterized by a straight line as shown in the image attached below, with the following ordered pairs (0, 0), (1, -2),(2, -4), and (-1, 2).
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Given the four points E(2,5), F(7,1), G(2,-3), and H(-8,5) is EF parallel to GH?
EF is parallel to GH by using the slope formula. Both the slope is equal to -4/5.
Given that the information below:
E(2,5), F(7,1), G(2.-3) and H(-8,5) is EF parallel to line GH.
By using slope formula method we can compare the slopes of line EF & line GH.
Where m = (y2 - y1 ) / (x2 - x1) slope of segment become (x1,y1) & (x2,y2)
EF Slope = (1-5)/(7-2) = -4/5
GH Slope = (5-(-3)/(-8-2) = 8/-10 = -4/5
By using slope formula method EF slope is -4/5 & GH slope is -4/5. Both the slopes of the equation is equal. In this way we can prove that the EF is parallel to GH.
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How do you convert 200 Kilogram (kg) to Pound (lb)?
The actual weight of a pound is 0.45359237 kilos.
How much in kilogrammes are converted to pounds?To Change the Price Per Pound (Pp) to the Price Per Kilogram (Kg), In order to convert cost per kilogramme to cost per lb, simply multiply the kg price by 2.2046 to get the lb price.In essence, one kilogramme is equivalent to two pounds (there is a longer decimal position, but I abbreviate it to 2.2046).In the fields of mathematics and engineering, converting from kilogrammes to pounds is a frequent operation; fortunately, it is also a simple one. For most conversions, all you need to do is multiply the number of kilogrammes by 2.2 to get the number of pounds.To learn more about Pound (lb) refer to:
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this is due tomorrow pls help
Note that the angle that is vertical to ∠AEB is ∠BEC.
What is a vertical angle?Vertical angles are the angles formed when two lines intersect. The term "vertical" in this context refers to the fact that they share the same Vertex (corner point), rather than the normal connotation of up-down.
Vertical angles of equal measure are always congruent. Both vertical angle pairings (four angles total) always add up to 360 degrees. Angles formed by each pair of vertical angles are referred to as neighboring angles, and they are supplementary (the angles sum up to 180 degrees).
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Lyla invested $140. She earned a simple interest of 3% per year on the initial investment. If no money was added or removed, what is the amount of money Lyla will earn at the end of two years?
Answer:
8.40
Step-by-step explanation:
Since, the simple interest is,
I= p x r x t
100
Where,
P=Principal
r= rate per periods
t= number of periods
Here, P=140, r=3%, t = 2,
the simple interest will be
I= 140 x 3 x 2
100
= 840
100
=8.40
I NEED HELP!!!!!!!!!
Note: Enter your answer and show all the steps that you use to solve this problem in the space provided. Find the missing values of the variables. The diagram is not to scale.
Answer:
y = 64
x = 99
Step-by-step explanation:
180 - 116 = y
y = 64
A parallelogram is 360 degrees.
360 - 72 -125 - 64 = x
x = 99
I hope this helps!
Find the distance between A and B. Round to the nearest hundredth.
A (-3,-6)
B (-6,1)
Answer: 10.30
Step-by-step explanation:
Distance formula = [tex]\sqrt{(x_{2} +x_{1})^2+(y_{2} +y_{1})^2}[/tex]
A ([tex]x_{1} ,y_{1}[/tex]) = A(-3,-6) so [tex]x_{1} = -2[/tex] and [tex]y_{1} =-6[/tex]
B ([tex]x_{2} ,y_{2}[/tex]) = B(-6, 1) so [tex]x_{2} = -6[/tex] and [tex]y_{2} = 1[/tex]
Plug in the values. You should get [tex]\sqrt{(-6 +-3)^2+(1 +-6)^2}=\sqrt{(-9)^2+(-5)^2}=\sqrt{81+25}=\sqrt{106}[/tex]
≈10.30
Rusty Reft, who lives in Territory 5, carries 10/20/5 compulsory liability insurance along with optional collision that has a $300 deductible. Rusty was at fault in an accident that caused $1,400 damage to the other auto and $3,100 damage to his own vehicle. Also, the courts awarded $12,800 and $9,200, respectively, to the two passengers in the other car for personal injuries.
a. How much will the insurance company pay?
b. What is Rusty’s share of the responsibility?
Answer:
A = $2,800 B = $12,600
Step-by-step explanation:
a. How much will the insurance company pay?
The liability insurance will cover the damages to the other car ($1,400) and the personal injury claims ($12,800 + $9,200) up to the limit of $10,000. So the insurance company will pay $10,000.
The collision insurance covers the damage to Rusty's own car, minus the $300 deductible, so the insurance company will pay $3,100 - $300 = $2,800.
Therefore, the insurance company will pay a total of $10,000 + $2,800 = $12,800.
b. What is Rusty’s share of the responsibility?
Rusty is responsible for paying the remaining amount not covered by insurance, which is $1,400 + $3,100 + $12,800 + $9,200 - $12,800 = $12,600.
Therefore, Rusty's share of the responsibility is $12,600.
evaluate the line integral where c is the straight line segment from point to point
The line integral [tex]\int_{c}xds[/tex]where c is the straight line segment from point (1, 2) to point (5, 10) is 12√5.
Using the two-point form of the line, we can compute the equation of the line connecting the points and obtain:
y = 2x
Consider x = t. Now the result is:
y = 2t
We can write C as:
[tex]\vec{r}(t)=t\hat{i}+2t\hat{j}[/tex]
Now, t ranges from t = 1 to t = 5 since x = t and x range from x = 1 to x = 5. When we differentiate the C equation, we obtain:
[tex]\vec{r}'(t)=\hat{i}+2\hat{j}[/tex]
Now finding the magnitude
[tex]|\vec{r}'(t)|=\sqrt{(1)^2+(2)^2}[/tex]
[tex]|\vec{r}'(t)|=\sqrt{1+4}[/tex]
[tex]|\vec{r}'(t)|=\sqrt{5}[/tex]
Now, the integral will be:
I = [tex]\int_{t=1}^5t\:(\sqrt 5)dt[/tex]
I = [tex]\sqrt {5}\int_{t=1}^5tdt[/tex]
Now integrating
I = [tex]\sqrt {5}\left(\frac{t^2}{2}\right)_{t=1}^{5}[/tex]
I = [tex]\sqrt {5}\left(\frac{5^2}{2}-\frac{1^2}{2}\right)[/tex]
I = √5(25/2 - 1/2)
I = √5(12.5 - 0.5)
I = 12√5
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The complete question is:
Evaluate the line integral [tex]\int_{c}xds[/tex]where c is the straight line segment from point (1, 2) to point (5, 10).
How to rotate a triangle 270 degrees counterclockwise?
Rotating a triangle 270 degrees counterclockwise is the same as rotating a figure 90 degrees clockwise. The solution has been obtained using the concept of rotation.
What is rotation?
A rotation in mathematics is a transformation that revolves a shape around a fixed point.
We have a specific rule that we can use to do this that is based on the fact that a 270° anticlockwise rotation is the same as a 90° clockwise rotation. One such rotation is to rotate a triangle 270° anticlockwise.
On a graph, we can rotate a triangle 270° anticlockwise by passing each of its vertices through the transformation shown below:
Change the x and y coordinates after multiplying the x coordinate by a negative number.
Consequently, (x,y) becomes (y,-x).
Hence, this way we can rotate a triangle 270 degrees counterclockwise.
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I dont understand may you please help me
Answer:66
Step-by-step explanation:
336666
How many ways can you split 12 people into 3 teams of 4?
By using permutations and combinations, We can arrange 12 people into 3 teams of 4 in 5775 different ways.
Here, we will use the concept of permutations and combinations to solve the question.
We have to arrange 12 people into 3 teams of 4.
We know that ⁿCr = n! / [(n - r)! × r!]
No. of ways to select the first 4 people in the first group = ¹²C₄
= 12! / [(12 - 4)! × 4!]
= 12! / [(8! × 4!)]
= 495
No. of ways to select 4 people from the remaining 8 for the second group = ⁸C₄ = 8! / [(8 - 4)! × 4!] = 8! / [(4! × 4!)] = 70
No. of ways to select 4 people from 4 for third group = ⁴C₄ = 4! / [(4 - 4)!] × 4!] = 4! / [(0! ×4!)] = 1
Total no. of ways to select people for group = (495 x 70 x 1) / 3! = 34650 /6 = 5775
Hence, we can split 12 people into 3 teams of 4 in 5775 different ways.
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A company model it net income, in thouand of dollar, with the function f(x) = 9x2 – 54x – 144, where x i the number of unit of it product old. How many unit of it product doe the company need to ell in order for the net income to equal $0?
The company needs to produce 8 units in order for the net income to equal $0
What is factoring?Is a technique that consist of decomposition of a factor into a product of another factor, which when multiplied together give the original number.
the equation given can be simplified as:
f(x) = 9x² - 54x - 144.
f(x) = 9(x² - 6x - 16)
Then factoring we have:
f(x) = 9[(x + 2)(x - 8)]
It has a net income equals to 0 when:
f(x) = 0, so the variables are:
x + 2 = 0 -> x = -2
x - 8 = 0 -> x = 8
As the amount of products sold have to be positive the amount of product is 8
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Small market orders copies of a certain magazine for its magazine rack each week. Let X = demand for the magazine, with the following pmf. X 1 2 3 4 5 6 p(x) 1 14 2 14 3 14 3 14 2 14 3 14 Suppose the store owner actually pays $2. 00 for each copy of the magazine and the price to customers is $4. 0. If magazines left at the end of the week have no salvage value, is it better to order three or four copies of the magazine? [Hint: For both three and four copies ordered, express net revenue as a function of demand X, and then compute the expected revenue. ] What is the expected profit if three magazines are ordered? (Round your answer to two decimal places. ) $ 27/7 Incorrect: Your answer is incorrect. What is the expected profit if four magazines are ordered? (Round your answer to two decimal places. ) $ How many magazines should the store owner order
Small market orders copies of a certain magazine for its magazine rack each week. The number of copies to order to generate more revenue is 4.
Let the profit y, the demand for the magazine be x , the number owner ordered k , then we get :
y = 4x - 2m ( m > x)
y = 2m ( m ≤ x)
If m = 3 we get
E (y) = 1/14 (4 - 2x3) + 2/14 (4x2 - 2x3) + 3/14 (2x3) + 3/14 (2x3) + 2/14 (2x3) + 3/14 (2x3)
E(y) = 68/14
If m = 4 get,
E (y) = 1/14 (4 - 2x4) + 2/14 (4x2 - 2x4) + 3/14 (4x3 -2x4) + 3/14 (2x4) + 2/14 (2x4) + 3/14 (2x4)
E(y) = 72/14
E (y)m=4 > E(y)m=3
So order number is 4.
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solve Edson exercises in sets that are each t minutes long. He does 6 sets of push-ups, 3 sets of pull-ups, and 4 sets of sit-ups. Use an expression to represent the time it takes Edson to exercise as the sum of three different terms. Simplify the expression. Enter your answer in the box.
The time taken by Edson to complete his workout routine is 13 t
What is the unitary method?The unitary method is a method in which you find the value of a unit and then the value of a required number of units.
Given here: Edson exercises in sets that are each t minutes long. He does 6 sets of push-ups, 3 sets of pull-ups, and 4 sets of sit-ups
Now Edson does 6 sets of pushups time taken to finish the sets are
Time ( pushups)=6t
similarly, the time it takes to complete the other exercises are
Time(pull ups)=3t
Time(sit ups)=4t
Thus total time taken is equal to 3t+4t+6t=13t
Hence, The time taken by Edson to complete his workout routine is 13 t
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Can anyone help me solve this?
On the right side, we can just simplify the expression to 1+tanθ+cotθ
The Identity equationYes, this is true.
The left side of the equation can be simplified to 1, which is equal to the right side.
Left side:
(cotθ/1-tanθ)+(tanθ/1-cotθ)
= (cotθ + tanθ)/(1 - tanθ - cotθ)
= (tanθ + cotθ)/(1 - tanθ - cotθ)
= 1/(1 - tanθ - cotθ)
= 1
Right side:
1 + tanθ + cotθ = 1
To prove that this equation is true for any value of θ, we must first expand and simplify each side of the equation. On the left side, we can express the fraction as a product of two terms, giving us (cotθ*(1-tanθ))+(tanθ*(1-cotθ)).We can then factor out cotθ and tanθ, giving us (cotθ*(1-tanθ+1-cotθ))+(tanθ*(1-cotθ+1-tanθ)).We can then combine the terms that are being multiplied by each side to get (cotθ+tanθ)*(1-tanθ-cotθ). Since the brackets are equal to 0, this simplifies to 0*(1-tanθ-cotθ), which is equal to 0.To learn more about Identity equation refer to:
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Answer:
See proof below
Step-by-step explanation:
(Please excuse any typos and point them out to me. It is a pain typing in LateX)
We are asked to prove:
[tex]\dfrac{\cot\theta}{1-\text{\ensuremath{\tan\theta}}}+\dfrac{\tan\theta}{1-\cot\theta}=1+\tan\theta+\cot\theta[/tex]
Since
[tex]\tan\theta =\dfrac {\sin\theta}{\cos\theta}\\\\\cot\theta = \dfrac {\cos\theta}{\sin\theta}\\\\[/tex]
the left side becomes
[tex]\dfrac{\cos\theta/\sin\theta}{1-\sin\theta/\cos\theta}+ \dfrac{{\sin\theta/\cos\theta}}{1-\cos\theta/\sin\theta} \cdots \cdots \cdots(1)[/tex]
For ease of expressing this let
[tex]a = \sin\theta\\b = \cos\theta\\[/tex]
Substituting the above in expression (1) gets us
[tex]\dfrac{b/a}{1- a/b} + \dfrac{a/b}{1-b/a}[/tex]
Let's take the individual terms and simplify each
[tex]\dfrac{b/a}{1- a/b} = \dfrac{b}{a}\cdot \dfrac{1}{1-\dfrac{a}{b}}[/tex]
[tex]\dfrac{1}{1-\dfrac{a}{b}}= \dfrac{1}{\dfrac{b-a}{b}}=\dfrac{b}{b-a}[/tex]
Therefore
[tex]\dfrac{b}{a}\cdot\dfrac{b}{b-a} = \dfrac{b^2}{a(b-a)}[/tex]
For the second term we get
[tex]\dfrac{a/b}{1-b/a} = \dfrac{a^2}{b(a-b)}[/tex]
Noting that (a - b) = - (b - a)
the entire expression becomes
[tex]\dfrac{b^2}{a(b-a)} = - \dfrac{a^2}{b(b-a)}[/tex]
Multiply this expression throughout by the LCM which is
[tex]ab(b-a)[/tex]
[tex]\dfrac{b^2 \cdot b - {a^2}\cdot a}{ab(b-a)}\\\\= \dfrac{b^3 - a^3}{ab(b-a)}\cdots\cdots(2)\\[/tex]
(b -a) cancels in numerator and denominator on both sides giving
For the numerator we have the identity
[tex]b^3 - a^3 = (b-a)(b^2 + ab + b^2)[/tex]
So the expression in (2) becomes
[tex]\dfrac{(b-a)(b^2 + ab + a^2)}{ab(b-a)}\\\\\textrm{(b-a) cancels from the numerator and denominator}\\\\[/tex]
This leaves us with:
[tex]\dfrac{(b^2 + ab + a^2)}{ab}[/tex]
Substituting for a = sinθ and b = cosθ we get
[tex]\dfrac{\cos^2\theta + \sin^2\theta+ \cos\theta\sin\theta}{\sin\theta\cos\theta}[/tex]
Divide each term in the numerator by [tex]\sin\theta \cos\theta[/tex]
==> [tex]\dfrac{cos^2\theta}{\sin\theta\cos\theta} + \dfrac{\sin^2\theta}{\sin\theta\cos\theta}+ \dfrac{\sin\theta\cos\theta}{\sin\theta\cos\theta}}[/tex]
==> [tex]\dfrac{\cos\theta}{\sin\theta} + \dfrac{\sin\theta}{\cos\theta} + 1[/tex]
[tex]\cot\theta + \tan\theta + 1\\== > 1 + \tan\theta + \cot\theta[/tex]
Hence PROVED
When u, v are nonzero vectors, then Span{u, v} contains only the line through u and the origin, and the line through v and the origin.
When u, v are nonzero vectors, then Span{u, v} contains only the line through u and the origin, and the line through v and the origin, this statement is false.
The span of two nonzero vectors u and v, Span{u, v}, is the set of all linear combinations of u and v. This set can include not only the lines through u and the origin, and the line through v and the origin, but also all other points on the plane defined by these two lines.
For example, if u = (1, 0) and v = (0, 1), then Span{u, v} would be the set of all points in the x-y plane, since every point in the plane can be represented as a linear combination of u and v.
Therefore, Span{u, v} contains not only the lines through u and the origin, and the line through v and the origin, but all other points on the plane defined by these two lines.
Correct Question :
When u, v are nonzero vectors, then Span{u, v} contains only the line through u and the origin, and the line through v and the origin. True or fasle.
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