Answers
The distance of the ship from lighthouse found using sine law at both locations are 14.12 miles and 4.8 miles.
What is sine law?
The sine law asserts that the ratio of a triangle's side length to the sine of its opposing angle, which is constant for all three sides.
From the given figure we can find θ = 30 - 10 = 20°
According to sine law, we can write the following equation,
[tex]\frac{9.6}{sin \theta} = \frac{x}{sin \angle ABL} = \frac{y}{sin \angle LAB} \\[/tex]
From the figure, we get that ∠LAB= 10° (alternate angles)
∠LBC = 30°
∠ABL = 180 - 30 = 150°
θ = 20°
Substituting above values in the equation we get ,
[tex]\frac{9.6}{sin 20} = \frac{x}{sin 150} = \frac{y}{sin 10} \\\\\frac{9.6}{0.34} = \frac{x}{0.5} = \frac{y}{0.17}[/tex]
Solving the above we get,
x = 14.12
y = 4.8
Therefore the ship began 14.12 miles from the lighthouse and after travelling 9.6 miles south it is at 4.8 miles from the lighthouse.
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Related Questions
I just need help with question A
Suppose your cousin missed class and asks you, “Why do we plug in 0 for x to find y-intercepts and 0 for y to find x-intercepts? That doesn’t make sense to me.” Explain to your cousin WHY this process works. Use the line
5x+4y=14 as an example.
A) A description in words why this works
Answers
The reason why the cousin's procedure for getting the x and y intercepts work is because they represent the definition of those points on a graph.
How to find the x and y intercepts?The x-intercept of a graph or function simply refers to the point at which the graph crosses the graph crosses the x-axis or the point at which y = 0. In a similar manner, the y-intercept simply refers to the point at which the graph crosses the y-axis or simply the point at which x = 0.
Now, we are given the equation of the line as;
5x + 4y = 14
The general form of equation of a line in slope intercept form is;
y = mx + c
where;
m is slope
c is y-intercept
Rewriting the equation of the line in slope intercept form is;
y = -⁵/₄x + ⁷/₂
Now, the reason why we plug in 0 for x to get the y-intercept is because when x = 0, that is only where we can get the y-intercept as the corresponding value of x will be traced to 0.
Similarly, the reason why we plug in 0 for y to get the x-intercept is because when y = 0, that is only where we can get the x-intercept as the corresponding value of y will be traced to 0.
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15) Find x if m
m/MTU = 38°, and m/STU = 2x + 88.
x + 50 = 2x + 88
Answers
Therefore , the solution of the given problem of equation comes out to be x = 38.
What is equation?Equivalent is the meaning of the Latin term equ. A lot of variable words in the English language, such as adequate, equator, and equality, have Latin roots as their root words. Since both side of an equations are, by definition, "equal" to one another, the Latin root term equ makes its way into our vocabulary quite simply.
Here,
Given : m m/MTU = 38°, and m/STU = 2x + 88.
thus,
x + 50 = 2x + 88
To find the value of x:
We do,
=> x + 50 = 2x +88
=> x -2x = 88 -50
=> -x = -38
=> x = 38
Therefore , the solution of the given problem of equation comes out to be x = 38.
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Complete the following ordered pairs
(x,y) so that they satisfy the equation
5x+3y=15.
(-6,3)
(3,0)
( , -10)
(, 10)
Answers
Answer:(0,5
Step-by-step explanation:
profits of calculess corporation this year were 140% of profits last year. if profits this year were $4900, what were profits last year.
Answers
The profits last year were $2041.66
What are percentages?A percentage is a number or ratio that can be expressed as a fraction of 100. If we have to calculate the percent of a number, divide the number by the whole and multiply by 100. Hence, the percentage means, a part per hundred. The word percent means per 100. It is represented by the symbol “%”
Given here, profits of calculess corporation this year were 140% of profits last year. if profits this year were $4900, then let the profit last year be x then we have 2.4x=4900
x=4900/2.4
x=2041.66
Hence, The profits last year were $2041.66
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Is this relation a function? Justify your answer.
11
10
9
8
7
654321
1 2 3 4 5 6 7 8 9 10 11
OA. Yes, because every x-value corresponds with a single y-value.
OB. Yes, because the number of x-values is the same as the number of
y-values.
OC. No, because two points with the same y-value have different x-
values.
OD. Yes, because every x- and y-value is positive.
Answers
Due to the fact that element 1 relates to two distinct images, namely 3 and 5. The relationship hence is not a function.
What is meant by function?A function is a relationship between a collection of inputs and one output per input. A function is, to put it simply, an association between inputs where each input is linked to exactly one output.For each function, a domain, codomain, or range exists. A function is often represented by the expression f(x), where x represents the input.A "chunk" of code that may be utilized repeatedly rather than having to be typed down repeatedly is called a function.Programmers can break a huge problem down into smaller components, each of which can perform a specific purpose, by using functions.Therefore,
(i) {(2,1),(5,1),(8,1),(11,1),(14,1),(17,1)
Every input has a single output, proving that it is a function.
The elements in the domain of the provided relation are therefore 2,5,8,11,14, and 17.
Here, domain is equal to two and range is one.
(ii) (2,1),(4,2),(6,3),(8,4),(10,5)(12,6),(14,7)
Every input has a single output, proving that it is a function.
Therefore, 2, 4, 6, 8, 10, 12, and 14 make up the domain of the provided relation.
Here, range is 1,2,3,4,5,6,7 and domain is 2,4,6,8,10,12,14.
(iii) (1,3),(1,5),(2,5)
Due to the fact that element 1 relates to two distinct images, namely 3 and 5. The relationship hence is not a function.
The complete question is:
Which of the following relations are functions? Give reasons.
If it is a function determine its domain and range
(i) (2,1),(5,1),(8,1),(11,1),(14,1),(17,1)
(ii) {(2,1),(4,2),(6,3),(8,4),(10,5),(12,6),(14,7)
(iii) {(1,3),(1,5),(2,5)
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simplify the expression 14+5(x+3) - 7x
Answers
Answer:
- 2x +29
Step-by-step explanation:
Open up the parenthesis, then simplify
14 + 5x + 15 - 7x
5x - 7x + 14 + 15
-2x + 29
[tex]14+5(x+3) - 7x[/tex]
Distribute:
[tex]14+(5)(x)+(5)(3)-7x[/tex]
[tex]14+5x+15-7x[/tex]
Combine Like Terms:
[tex](5x-7x)+(14+15)[/tex]
[tex]\fbox{-2x + 29}[/tex]
his mapping shows a functional relationship. A mapping diagram shows a relation, using arrows, between domain and range for the following ordered pairs: (2, 4), (negative 3, 0), (negative 1, negative 1), (4, 3). When f(x)=4, what is the value of x? 0 2 3 4
Answers
when f(x)=4, then the value of x is 2
What is a function?A relation is a function if it has only One y-value for each x-value.
Given that A mapping diagram shows a relation, using arrows, between domain and range for the following ordered pairs:
The ordered pairs are (2, 4), (-3, 0), (- 1, -1), (4, 3)
We need to find value of x when f(x)=4
In the (x, y) ordered pair the x place represents the domain value and y place represents the range.
In the order pairs we have to look for the y value which has 4.
The corresponding element of 4 is the x value
So the value of x is 2 if f(x)=4
Hence, when f(x)=4, then the value of x is 2
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4. Find the sum of the first 21 terms in a series where a = 20 and t21= 400.
5. Find the sum of the first 100 terms in a series where a = 0 and t100= 99.
6. Find the sum of the first 50 terms in a series where a = 4 and t50= 196.
7. Find the sum of the first 100 terms in a series where a = 0 and t100= 99.
8. Find the sum of the first 10 terms in an arithmetic series where t7= 7 and t10= 13.
9. Find the sum of the first 10 terms in an arithmetic series where t5= 15 and t10= 45.
10. Find the sum of the first 30 positive multiples of 5.
Answers
The first 100 terms of a series add up to 4950, and the first 21 terms of a series where a = 20 and t21 = 400 add up to 4410
what is arithmetic progression ?An arithmetic progression is when the difference between each phrase that follows another in a sequence is always the same. For instance, the sequence 5, 7, 9, 11, 13, and 15 is an example of an arithmetic progression with a 2 tolerance. A progression having a set tolerance between any two consecutive numbers is known as a "arithmetic progression" (A.P.). Two alternative types of mathematical progression exist: finite-length mathematical series A series that has a limited number of terms is a finite geometric progression. One may calculate the early, late, tolerance, and number of terms in a series using the terms in the series.
given
1) n6 = 6 where a = 5 and d = 5 = 105
2)n6 =6 where a = 9 and d = 12 = 234
3) n5= 5 where a = 5.7 and d = 1.4 = 42.5
4) a = 20 and t₂₁ = 400 ,
= 380
Therefore, Sum of 21 terms = 4410
5) In series a = 0 and t₁₀₀ = 99 = 4950
The first 100 terms of a series add up to 4950, and the first 21 terms of a series where a = 20 and t21 = 400 add up to 4410
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Write an equation
Y=|x| 10 units down
Answers
To translate the point P(x,y) , a units left and b units down, use P'(x−a,y−b) .
Write an equation Y=|x| 10 units down?y=|x-10|-3
Inside, is opposite. So therefore, to move it right 10 units, you subtract ten. However, outside of the absolute value, it is normal, so to move it down 3 units, you subtract 3.The number –45.3 is 45.3 units away from zero and the number 10 is 10 units away from zero. Therefore, the absolute value of any number is really a positive number (or zero).As we know that absolute value means remove the negative sign from the number and make it positive. So we get the absolute value of -10 as 10. So we get the absolute value of 10 as 10. Hence absolute value of -10 and 10 is 10 and 10 respectivelyTo know more about units visit:
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(4 x 5) + ( 4 x 2) = 4 x (5 + n)
Answers
Answer:
Step-by-step explanation:
1. (4x5) is 20
2. (4x 2) is 8
3. 20 plus 8 is 28
4. 28= 4x (5+n)
5. isolate n by dividing 4 on both sides= 28/4= 4+n
6. subtract 4 on both sides to get 7-4=n
7. n=3
A concave lear of focal length of 20cm forms an image 1/2 the size of the object .The object distance?
Answers
Answer:
60cm
Step-by-step explanation:
given:
focal length (f) = 20cm
image height (v) = ½u
object height (u) = ?
(1÷v) + (1÷u) = (1÷20)
(1÷(½u)) + (1÷u) = (1÷20)
(2÷u) + (1÷u) = (1÷20)
(3÷u) = (1÷20)
u = 20 × 3
u = 60cm
Bethany and Samuel are buying a new house. They have $25,000 saved for a down payment and know that they can afford a monthly payment of $1,500 or less. They also know that the best interest rate they can get is 5.1% annually and they want to sign a 30-year mortgage.
1. This equation is used to find the monthly payment, m, given the monthly interest rate, i, the principal, P, and the number of interest periods, n, in months:
Using the equation, find the largest possible principal P for their situation. In other words, what is the largest amount of money they can borrow? (Hint: To answer the question, you must rearrange the equation.)
Answers
The largest amount of money they can borrow is $$276.268.65
How to calculate the largest amount?The main formula used in this problem is the Present Value of an annuity Formula:
P = R[1-(1+i)⁻ⁿ]/i
Where;
P: present value (the mortgage value)R: periodic paymentt: time (in years)r: annual interest ratem: number of compounding per yeari=r/m (rate per period)n=mt the total number of paymentsThe maximal monthly payment that Bethany and Samuel can afford is $1,500.
The best interest rate is %5.1 annually, and they want to sign a 30-year mortgage.
This implies that
R=1,500
t=30
r=%5.1 = 5.1 /100 = 0.051
m=12, because they make the payments monthly
i=r/m=0.051/12=0.00425
n=mt = 12*30=360
That is,
i) they will make 360 monthly payments,
ii) the largest amount of money they can borrow can be found using the formula:
P = R[1-(1+i)⁻ⁿ]/i
Substituting the known values:
P = 1500[1-(1+0.00425)⁻³⁶⁰]/0.00425
Therefore, $276.268.65 is the largest possible principal for the situation
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Write three ratios that are equivalent to 75% and three
ratios that are equivalent to 100%.
Answers
The three ratios that are equivalent to 75% are 75/100, 15/20 and 3/4 and the three ratios that are equivalent to 100% are 100/100, 20/20, and 4/4.
What is ratio?
A ratio in mathematics demonstrates how many times one number is present in another. For instance, if a bowl of fruit contains eight oranges and six lemons, the ratio of oranges to lemons is eight to six. The ratio of oranges to the total amount of fruit is 8:14, and the ratio of lemons to oranges is 6:8.
First we have to write the equivalent ratios for 75%.
Since, 75% can be written as,
75/100
Simplifying this, 15/20
Again simplifying this, 3/4
Hence, the three ratios that are equivalent to 75% are 75/100, 15/20 and 3/4.
Now to write the equivalent ratios for 100%.
Since, 100% can be written as,
100/100
Simplifying this, 20/20
Again simplifying this, 4/4
Hence, the three ratios that are equivalent to 100% are 100/100, 20/20, and 4/4.
Therefore, the three ratios that are equivalent to 75% are 75/100, 15/20 and 3/4 and the three ratios that are equivalent to 100% are 100/100, 20/20, and 4/4.
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Complete the square for the expression. Then factor the trinomial.
x2 - 6x
Expression:
Factored form:
Answers
Expression: x² - 6x + 9
Factored form: (x - 3)²
Using the concept of factorization.
What is factorization?The breaking or breakdown of an entity (such as an integer, a matrix, or even a polynomial) into such a combination of some other unit, or factors, whose multiplication yields the original number, matrix, etc., is known as factorization or factoring in mathematics.
Similarly to this, factoring is a very useful technique in algebra that is utilized at all levels. Additionally, to, of course, simplify complex expressions, it offers a standard way for solving quadratic equations. Graphing functions also makes advantage of it.
Factor the expression, put each factor equal to 0, and then solve each equation to solve a quadratic equation by factoring. To do this, rearrange all terms on one side of the equation such the other side equals 0.
Given that,
x² - 6x
Now, to find the value of c = (b/2)²
divide the coefficient of x by 2.
So, divide 6 by 2 we get: ( -1.3)²
Next, multiply -1 by 3: (-3)²
Now, raise -3 to the power of 2, we get: 9
So, the expression becomes: x² - 6x + 9
Factorized form: (x - 3) (x -3)
or we can write (x - 3)²
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Suppose we want to choose 6 letters with out replacement from 13 distinct letters. How many ways can this be done if the order of the choices is not relevant? How many ways can this be done if the order of the choices is relevant?
Answers
Suppose we want to choose 6 letters with out replacement from 13 distinct letters.
a. The number of ways this be done if the order of the choices is not relevant is 1,235,520 ways.
b. The number of ways this be done if the order of the choices is relevant is 1716 ways.
How to find the number of ways?a. Number of ways if choice are not relevant
[tex]^nP_r = n!/ (n-r)![/tex]
Where:
n =13
r = 6
So,
[tex]^1^3P_6[/tex] = 13! / (13 - 6)!
[tex]^1^3P_6[/tex] =6,227,020,800 /7!
[tex]^1^3P_6[/tex] =6,227,020,800/5,040
[tex]^1^3P_6[/tex] = 1,235,520 ways
b. Number of ways if choice are relevant
[tex]^nC_r =n!/ r! (n-r)![/tex]
[tex]^1^3C_6[/tex] = 13! / 6!(13-6)!
[tex]^1^3C_6[/tex] = 13! / 6! × 7!
[tex]^1^3C_6[/tex] =6,227,020,800 / 720×5040
[tex]^1^3C_6[/tex] =6,227,020,800 / 3,628,800
[tex]^1^3C_6[/tex] = 1,716 ways
Therefore the number of ways are 1,235,520 ways and 1,716 ways.
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1 yard in 6 minutes
Question 1
Part A
Find the unit rate.
Enter the correct answer in the box.
Answers
Answer: 0.166666667 yards OR 0.1524 meters OR 0.5 feet
Step-by-step explanation:
1 / 6 = 0.166666667 yards
1 yard = 0.9144 meters
0.9144 / 6 = 0.1524 meters
1 yard = 3 feet
3 / 6 = 0.5 feet
The dollar value v (t) of a certain car model that is t years old is given by the following exponential function. v (t) = 26,000 (0.84)¹ Find the value of the car after 7 years and after 12 years. Round your answers to the nearest dollar as necessary. Value after 7 years: Value after 12 years: $0
Answers
I think it's 84 I'm not good with math though I multipled 7 and 12
2 alien spaceships are traveling toward each other from space stations that are 7100 km apart at speed of 110000 km
Answers
It would take 116.18 seconds for the space stations that are 7100 km apart at speed of 110000 km to meet
How to solve an equation?An equation is an expression that shows the relationship between two or more numbers.
The speed of an object is the ratio of the total distance travelled to the total time taken. It is given by:
Speed = distance / time
Distance = Speed * time
The spaceships are moving at 110000 km/h
The distance covered by each in x hours is:
Distance = 110000 * x = 110000x
Since they are 7100 km apart, hence:
110000x + 110000x = 7100
220000x = 7100
x = 0.03227 hours = 116.18 seconds
It would take 116.18 seconds for the spaceships to meet.
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$640 , 3% , 2 years Find the simple interest earned to the nearest cent for each principal , interest rate , and time
Answers
The simple interest for principal of 640 and rate of 3% and time of 2 years is $3.84
How to find the simple interestThe Simple interest the end of 2 years is calculated using the simple interest formula
The formula is stated as
Simple interest = Principal * time * rate / 100
In the problem the give data include
principal = $640
rate= 3%
time = 2 years
plugging in the values
Simple interest = Principal * time * rate / 100
Simple interest = 640 * 2 * 3 /100
Simple interest = 3840 / 100
Simple interest = $3.84
The simple interest is $3.84
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You're fitting a pipe into an existing fitting. The fitting allows a diameter variance of 0.001". If the interior diameter of the pipe is 0.49" and the pipe thickness is 0.005", What's the maximum allowable pipe diameter? a. 0.496, b.0.501, c. 0.499, d. 0.541
Answers
The maximum allowable pipe diameter is given as : 0.501. Option B
How to solve for the diameterFrom the question the interior diameter of the pipe is said to be 0.49
The pipe is said to have the thickness of 0.005 inches.
The exterior diameter of the pipe would be solved as 0.49 + 2 * 0.005
= 0.49 + 0.01
= 0.5
The questions said the variance in the diameter of this pipe is 0.001
Then the maximum allowable pipe diameter would be solved as: 0.001 + 0.5 = 0.501
The diameter of the pipe that is allowable is 0.501
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A boat is purchased for $51,000 and has a resale value of $42,000 after a useful life of 6 years. Using a book value depreciation chart, calculate the number of years until the book value of the boat is $48,000.
Answers
The number of years until the book value of the boat is $ 48, 000, can be found to be 2 years
How to find the number of years ?To find the number of years till the book value of the boat is $ 48, 000, you first need to find the depreciation rate per year by the formula :
= ( Purchase price of boat - Resale value ) / Useful life
= ( 51, 000 - 42, 000 ) / 6
= 9, 000 / 6 years
= $ 1, 500 per year
In the first year, the value of the boat would depreciate to:
= 51, 000 - 1, 500
= $ 49, 500
In the second year, the value would depreciate to:
= 49, 500 - 1, 500
= $ 48, 000
It therefore takes 2 years for the book value to be $ 48, 000.
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Ifn(r'ns')+ n(r'ns)=3 n(r n s) = 4 & n(s'nr)= 7 find n(u).
Answers
Answer:
14
Step-by-step explanation:
You want to know the size of the universal set if ...
n(r'∩s') +n(r'∩s) = 3n(r∩s) = 4n(s'∩r) = 7UniverseThe given sets are mutually exclusive and exhaustive, so the sum of their sizes is the size of the universal set:
n(u) = 3 + 4 + 7
n(u) = 14
__
Additional comment
(r'∩s') ∪ (r'∩s) = r'∩(s' ∪ s) = r' . . . . n(r') = 3
(r∩s) ∪ (r∩s') = r∩(s ∪ s') = r . . . . . . n(r) = 4 +7 = 11
Then r' ∪ r = u, the universal set. . . . . n(u) = 3 +11 = 14
if june 13 was on monday what day will it be on june 13 math
Answers
Answer:
Monday??????
SteP-by-step explanation:
Terrence made a scale drawing of a boarding school. The scale he used was
7 centimeters: 3 meters. If the actual length of a building at the school is 300 meters, how
on is the building in the drawing?
Answers
The building in the drawing is 700 centimeters or 7 meters.
Define the unitary method.The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.
What are ratios and proportions?An ordered pair of integers a and b, represented as a / b, is a ratio if b is not equal to 0. A proportion is an equation that sets two ratios at the same value. For instance, you might express the ratio as follows: 1: 3 if there is 1 boy and 3 girls (for every one boy there are 3 girls) There are 1 in 4 boys and 3 in 4 girls.
Given 3 meters on real is equal to 7 centimeters on drawing
i.e. 3 meters=7 centimeters
or, 1 meter = 7/3 centimeters
We need to find reading for 300 meters so,
or, 300 meters=7/3*300 centimeters
=700 centimeters or 7 meters
Therefore the building in the drawing is 700 centimeters or 7 meters.
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A two story building measures 21 ft from the ground. A scale model shows the building to be 3 inches taller. How many inches would an 84 foot building be in the model?
Answers
Answer:
The scale of the model is (21 ft / 3 in) = 7 ft / 1 in. So an 84 foot building would be (84 ft / 7 ft) * 1 in = 12 inches tall in the model.
Step-by-step explanation:
hope it helps
Answer:
588 inches
Step-by-step explanation:
First, find the scale factor of the model by dividing the height of the real building by the height of the model: 21 ft / 3 inches = 21/3 = 7
7 inches/ft
Then, multiply the scale factor by the height of the 84 foot building to find the height of the model: 7 inches/ft * 84 ft = 7*84 = 588
588 inches
Tickets to a basketball game can be ordered online for a set price per ticket plus a $5.50 service fee. The total cost in dollars for ordering 5 tickets is $108.00. Which linear function represents c, the total cost, when x tickets are ordered? (A service fee is a single fee applied to the total, no matter the number of tickets purchased).
Answers
Considering the definition of a line, the linear function that represents the total cost when x tickets are ordered is c= 20.5x + 5.50.
What is a linear equationA linear equation o line can be expressed in the form y = mx + b
where
x and y are coordinates of a point.m is the slope.b is the ordinate to the origin and represents the coordinate of the point where the line crosses the y axis.Total cost in this caseIn this case, you know:
Tickets to a basketball game can be ordered online for a set price per ticket plus a $5.50 service fee. This is, as service fee is a single fee applied to the total, no matter the number of tickets purchased, the value of "b" is 5.50.The total cost in dollars for ordering 5 tickets is $108.00Being Total cost= set price per ticket× amount of tickets + 5.50, then:
108=set price per ticket× 5 + 5.50
108 - 5.50= set price per ticket× 5
102.5= set price per ticket× 5
102.5÷5= set price per ticket
20.5= set price per ticket
Finally, the total cost is Total cost= 20.5× amount of tickets + 5.50.
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Answer:
Its letter A
Step-by-step explanation:
Find the trigonometric ratios for both acute angles in fraction and decimal form to the nearest ten-thousandths when appropriate.
Answers
The trigonometric functions are sin A = 12/13, sin B = 5/13, cos A = 5/13, cos B = 12/13, tan A = 12/5, and tan B = 5/12 as fractions.
What are Trigonometric functions?Trigonometric functions are defined as the functions which show the relationship between the angle and sides of a right-angled triangle.
The trigonometric functions are given in the question
Apply the sine ratio on the given right triangle
⇒ sin(θ) = Perpendicular/hypotenuse
sin A = BC/AB
sin A = 12/13
sin B = AC/AB
sin B = 5/13
Apply the cosine ratio on the given right triangle
⇒ cos(θ) = Base/hypotenuse
cos A = AC/AB
cos A = 5/13
cos B = BC/AB
cos B = 12/13
Apply the tangent ratio on the given right triangle
⇒ sin(θ) = Perpendicular/Base
tan A = BC/AC
tan A = 12/5
tan B = AC/BC
tan B = 5/12
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Guys, how many triangles are in this picture? Triangles with stripes inside also count
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Therefore , as a result Two of a triangle's sides and the included SAS angle are said to be congruent if they match the corresponding sides and angles of another triangle.
What is the SAS?The Side-Angle-Side rule is applied to check the congruence of a set of triangles. The SAS rule states that a triangle is congruent if its two sides and included angles match those of another triangle.
Here,
if the first triangle's two sides and one of its included angles are equal to the second triangle's sides and one of its included angles.
Consequently, the SAS declares two triangles to be congruent. Two triangles are said to be congruent if their matching two sides and their included angles are found in a single triangle, according to the SAS theorem of congruence.
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1. Let
U = {(x, y, z) = R³ | x+y=2z = 0 = x - y},
and let
V = span {(1, 0, -1), (3, 1, 2)}.
Determine the dimensions of both U and V, proving your results.
Note that to prove the dimension, you should use the definition, which
means you need to find a basis. It is not enough to just state a basis,
you must explain why it is a basis.
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The dimension of V is 2, and it is spanned by the basis vectors (1, 0, -1), (3, 1, 2).
The dimension of a vector space is the number of vectors in a basis for the space. To find a basis for a subspace of R³, we need to find a set of linearly independent vectors that span the subspace.
First, let's consider U. The subspace U is defined by the equations x + y = 2z and x - y = 0.
Solving these equations for x and y, we can express them in terms of z: x = 2z and y = -2z. Therefore, every vector in U can be written as (x, y, z) = (2z, -2z, z) = z(2, -2, 1).
Since z is a scalar, the vector (2, -2, 1) is the only vector needed to span the subspace U, which means that it is a basis for U.
Therefore, the dimension of U is 1, and it is spanned by the basis vector (2, -2, 1).
Now let's consider V. The subspace V is defined by the set of vectors that can be written as a linear combination of the vectors (1, 0, -1) and (3, 1, 2). We can start by showing that these vectors are linearly independent. Suppose that there are scalars a and b such that a(1, 0, -1) + b(3, 1, 2) = (0, 0, 0). Then we have:
a + 3b = 0
b = 0
-a = 0
Since a and b are real numbers, the above equation only holds if a = 0 and b = 0. This shows that (1, 0, -1) and (3, 1, 2) are linearly independent, and a basis of V.
Therefore, the size of V is 2, and it is traversed by the basis vectors (1, 0, -1), (3, 1, 2).
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A business gained 50 new customers in April, 100 new customers in May, and 150 new customers in June. During that time, they were running a campaign with a budget of $3000. What was their Customer Acquisition cost?
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Their Customer Acquisition cost was $10
What is customer acquisition cost?Customer Acquisition Cost is the cost of winning a customer to purchase a product or service. As an important unit economic, customer acquisition costs are often related to customer lifetime value. Customer acquisition cost can also be referred as the fee associated with convincing a customer to buy your product or service.
To calculate customer acquisition cost
Customer Acquisition Cost (CAC) = Customer Acquisition Cost/total marketing cost
From the question,
The business gained 50+100+150= 300 new customers
Campaign budget = $30000
Customer Acquisition Cost (CAC) = $3000/300
=$10
Hence, it cost the business $10 to acquire a new customer
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