Yes, it is possible to have a graph of a function that is symmetric about both the y-axis and the origin. This means that if (x, y) is a point on the graph, then (-x, y) and (x, -y) are also points on the graph.
In other words, the function has both reflectional symmetry about the y-axis and rotational symmetry about the origin. An example of such a function is y = x², which is a parabolic function that opens upwards and is symmetric about both the y-axis and the origin.
What is the function about?A graph is symmetric about the y-axis if the graph is unchanged when reflected over the y-axis. This means that for any point (x,y) on the graph, the corresponding point (-x,-y) will also be on the graph. A graph is symmetric about the origin if the graph is unchanged when reflected over the origin.
Therefore, This means that for any point (x,y) on the graph, the corresponding point (-x,y) will also be on the graph. When a graph is symmetric about both the y-axis and the origin, it means that the graph is unchanged when reflected over either the y-axis or the origin.
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A flyer posted at a market is a rectangular piece of paper that is 19 inches (in) long and 15 in. Wide. Find the area of the rectangle defined by this flyer
Please assist with the math question in the picture
PLEASE HELP 20 POINTS!
A new cruise ship line has just launched 3 new ships: the Pacific Paradise, the Caribbean Paradise, and the Mediterranean ParadiseThe Caribbean Paradise has 16 more deluxe staterooms than the Pacific Paradise. The Mediterranean Paradise has 28 fewer deluxe staterooms than four times the number of deluxe staterooms on the Pacific Paradise Find the number of deluxe staterooms for each of the ships if the total number of deluxe staterooms for the three ships is 1176
ANSWERS MUST ADD UP TO 1176
Answer:
Pacific Paradise = 198 deluxe stateroomsCaribbean Paradise = 214 deluxe stateroomsMediterranean Paradise = 764 deluxe stateroomsCheck: 198+214+764 = 1176 total rooms
==========================================================
Explanation
I'll define these variables:
p = number of deluxe staterooms on Pacific Paradisec = number of deluxe staterooms on Caribbean Paradisem = number of staterooms on Mediterranean ParadiseWe have these given facts:
The Caribbean Paradise has 16 more deluxe staterooms than the Pacific Paradise. The Mediterranean Paradise has 28 fewer deluxe staterooms than four times the number of deluxe staterooms on the Pacific Paradise The total number of deluxe staterooms for the three ships is 1176Fact 1 allows us to form the equation c = p+16
Fact 2 allows us to form the equation m = 4p-28
Fact 3 leads to the equation p+c+m = 1176
Let me know if you have any questions about how I formed the equations.
---------------------
The system of equations is:
[tex]\begin{cases}c = p+16\\m = 4p-28\\p+c+m = 1176\\\end{cases}[/tex]
Notice how the first two equations have the variable p on the right hand side, and no other variable. We'll use this to apply substitution and solve for p.
p+c+m = 1176
p+(p+16)+m = 1176 ... replace c with p+16
p+(p+16)+(4p-28) = 1176 ... replace m with 4p-28
6p-12 = 1176
6p = 1176+12
6p = 1188
p = 1188/6
p = 198
Use this value of p to find the other variables.
c = p+16 = 198+16 = 214
m = 4p-28 = 4*198-28 = 764
---------------------
Summary:
We found that p = 198, c = 214, and m = 764.
Pacific Paradise = 198 deluxe stateroomsCaribbean Paradise = 214 deluxe stateroomsMediterranean Paradise = 764 deluxe stateroomsAs a check,
198+214+764 = 1176 total rooms
The answer is confirmed.
Members of the city council want to know if a majority of city residents support a 1% increase in the sales tax to fund road repairs. To investigate, they survey a random sample of 300 city residents and use the results to test the following hypotheses:
H0:pH????:p=0.50>0.50
where p is the proportion of all city residents who support a 1% increase in the sales tax to fund road repairs. Given that P-value = 0.18, which of the following would be the most appropriate conclusion?
A. Because the P-value is large, we reject H0. We have convincing evidence that at most 50% of city residents support the tax increase.
B. Because the P-value is large, we fail to reject H0. We have convincing evidence that more than 50% of city residents support the tax increase.
C. Because the P-value is large, we fail to reject H0. We do not have convincing evidence that more than 50% of city residents support the tax increase.
D. Because the P-value is large, we reject H0. We have convincing evidence that at most 50% of city residents support the tax increase.
For P - value = 0.18, there is not sufficient evidence that more than 50% of city residents supports the tax increase. So, the right answer of this problem is option ( c ).
Hypothesis testing is a statistical method utilized to make a decision about a population, with results from a set of sampled data from said population. We consider the assumptions as the null hypothesis, or H and alternative hypothesis, that is, the hypothesis we wish to examine whether it was true or not. Using the sampled data, we can calculate the test statistic, and obtained what is known as the P-value. The P-value is used to draw a convenient vasis conclusions in hypothesis-testing and is often referred to as the observed significance level of the test. We have, survey as a random sample of 300 city residents and use the results to test the following hypotheses:
H₀ : p = 0.50
Hₐ : p>0.50
where p is proportion of all city residents who support a 1% increase in the sales tax to fund road repairs and p-hat ( sample proportion) = 158/300 = 0.52 , P - value = 0.18. Now, we check the options and try to draw the conclusion. P-value is the probability where the population proportion (p) is more extreme than sample proportion ( p-hat). but p-hat > p , i.e, P value is large, we fail to reject null hypothesis. So, There is not sufficient evidence to support null hypothesis. The option (c) is correct.
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2x + 2x + 2 = 4x + 2
one solution
no solutions
all real solutions
0 x = 8
HELP
Answer:
no solution
Step-by-step explanation:
2x+2x+2=4x+2
4x+2=4x+2
4x-4x=2-2
0=0
Answer:
x=8
Step-by-step explanation:
34=34, always true
Pls answer will reward
The f(8) of the exponential function is 131072
How to find f(8) of the exponential function?
The general form of the exponential function is f(x) = abˣ
First, derive the function using the table. That is:
From the table, when x = 0, f(x) = 2. Substitute the values into the function:
f(x) = abˣ
2 = ab⁰ (Note: b⁰ = 1)
2 = a(1)
2 = a
a = 2
Also, from the table, when x = 3, f(x) = 128. Substitute the values into the function:
128 = 2 × b³
128 = 2b³
64 = b³
b³ = 64
b = ∛64
b = 4
We can now write the exponential function for the data:
f(x) = abˣ
f(x) = 2(4)ˣ
Thus, f(8) = 2(4)⁸ = 131072
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1/15 results in a repeating decimal.
Answer:
below
Step-by-step explanation:
1/15 = 1 divided by 15 = .06 (underlined part repeats)
You are at a restaurant and your bill comes to $34.78. You need to add the sales tax which
is 6%. You have one $20 bill, two $10 bills, one $5 bill, five $1 bills, and some pennies
How much is your total bill including tax?
How much should you give the server so you do not get back any pennies?
What denominations will you use for this?
Use the equation editor to show your work.
The total bill is $36.87, which you can pay using one $20 bill, one $10 bill, one $5 bill, one $1 bill, and 0.87 in pennies to avoid getting any pennies back.
How much is the total bill?Let's calculate and add the 6% (sales tax)
[tex]34.78 x 0.06 = 2.08\\\\34.78 + 2.08 = 36.86[/tex]
This can be rounded to $36.87.
How much should you give the server and what denominations can you use to avoid getting pennies back?Considering the available bills, the best option to pay without getting pennies is :
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It is day 8 and you are getting hungry. Since hunting/fishing have not been successful so far, you decide to eat ants for protein. 100 grams of ants gives enough protein as an egg. How many ants is this if one ant is between 1 - 5 mg? (Give a range of values)
The range of ants to eat is 20,000-1,00,000 ants.
What is range?The difference between the highest and lowest values for a particular data collection is the range in statistics. For instance, the range will be 10 - 2 = 8 if the provided data set is 2, 5, 8, 10, and 3.
As a result, the range may alternatively be thought of as the distance between the highest and lowest observation. The range of observation is the name given to the outcome. Statistics' range reflects the variety of observations.
The answer is given below:
If each ant is of 1 mg:
1g = 1000 mg = 1000 ants
100g = 100000 mg = 100000 ants
=> Maximum 105 ants
Now if one ant is of 5 mg:
1g = 1000mg = 1000/5 = 200 ants
100g = 20000 ants
Minimum 20,000 ants
Hence the range of ants to eat is 20,000-1,00,000 ants.
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Using technology, find the measures of central tendency for the following raw quantitative data set. Round the answers to two decimal places. 59.8 61.4 69 71.2 78.7 83.5 89.6 94.1 84 75.2 79.7 83.6 74.8 74 83.3 73.8 90 68.3 71.7 74.2 69.3 68.5 71.3 78.5 73.2 91.4 78.7 56.4 73 65.6 69.9 73.8 80.1 78.7 75.8 66.7 79.7 66 73 85.3 72.1 78.5 79.9 75.6 71.3 69.6 58.6 64.3 63.2 79.9 70.1 80.4 88.2 79.4 74.6 65.2 70.6 77 73.2 81.5 83.3 77 82 78.5 75.4 78.5 73.6 86.8 74.4 65.2 mean - 85.16 Enter an integer or decimal number (more.. median - 35.5 I
The median of the data set is 75.80, rounded to two decimal places.
The mean of the data set can be calculated by adding up all the values and dividing by the number of values.
The formula is:
mean = (sum of all values) / (number of values)
For the given data set, mean = (59.8 + 61.4 + 69 + 71.2 + 78.7 + 83.5 + 89.6 + 94.1 + 84 + 75.2 + 79.7 + 83.6 + 74.8 + 74 + 83.3 + 73.8 + 90 + 68.3 + 71.7 + 74.2 + 69.3 + 68.5 + 71.3 + 78.5 + 73.2 + 91.4 + 78.7 + 56.4 + 73 + 65.6 + 69.9 + 73.8 + 80.1 + 78.7 + 75.8 + 66.7 + 79.7 + 66 + 73 + 85.3 + 72.1 + 78.5 + 79.9 + 75.6 + 71.3 + 69.6 + 58.6 + 64.3 + 63.2 + 79.9 + 70.1 + 80.4 + 88.2 + 79.4 + 74.6 + 65.2 + 70.6 + 77 + 73.2 + 81.5 + 83.3 + 77 + 82 + 78.5 + 75.4 + 78.5 + 73.6 + 86.8 + 74.4 + 65.2) / 60 = 6288.6 / 60 = 104.81
So, the mean of the data set is 104.81, rounded to two decimal places.
The median of the data set can be calculated by arranging the values in ascending order and finding the middle value. If there are an odd number of values, the median is the middle value. If there are an even number of values, the median is the average of the two middle values.
For the given data set, the median can be found as follows:
56.4 58.6 63.2 64.3 65.2 65.6 66 66.7 68.3 68.5 69 69.3 69.6 69.9 70.1 70.6 71 71.3 71.7 73 73 73.2 73.6 73.8 74 74 74.2 74.4 74.6 75.2 75.4 75.6 75.8 76.7 77 78 78.5 78.5 78.5 78.7 79 79.4 79.7 79.7 79.9 80.1 80.4 81.5 82 83 83.3 83.3 83.5 83.6 84 85.3 86.8 89.6 90 91.4 94.1
The median is 75.8.
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7,11,2,18,-7
What's the next number in this sequence and how did you arrive at that number
The last term of the missing pattern is 57.
What is Sequence?A sequence is a list of items where repetitions are allowed and hierarchy is crucial. The same as a set, it has members. The number of items determines how long the sequence is.
Given:
Sequence: 7,11,2,18,-7
As, the difference between the first two-term is 4,
while in the next two-term is 16(4²),
So, in the next two-term the difference will be equal to 64(4³).
Then, the last term is
x- (-7)= 64
x+ 7= 64
x= 57
Hence, the last term is 54.
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in the diagrams below parallel lines I and m are crossed by transversal n at an 108 degree angle as shown. which set of angles are supplementary to the 108° angle?
Our assumption that m II is intersected by a transversal n is wrong, thus none of the set of angles are supplementary to 108°.
What is transversal?A transversal in geometry is a line that intersects two other lines in the same plane at two different locations. It is possible to determine whether two or more other lines in the Euclidean plane are parallel by using transversals.
A line that crosses two lines in the same plane at two different geometric locations is referred to as a transversal. The angles that result from a transversal crossing of two lines in pairs can be successive internal angles, matching angles, or alternate angles.
The transversal angles. There are numerous angles created when a transversal splits two parallel lines. They are referred to as transversal angles.
Given, ∠1 = 70°
∠7 = 80°
Vertically opposite angles are equal.
∠1 = ∠3
⇒ ∠1 = ∠3 = 70°
As per the diagram, line 1 is intersected by line n.
Vertically opposite angles are equal.
∠5 = ∠7
=> ∠5 = ∠7 = 80°
Let m II l is intersected by a transversals n.
Then the pair of corresponding angles should be equal. i.e.,
∠1 = ∠5
= ∠3 = ∠7
but ∠1 = 70
and ∠5 = 80°
Therefore, our assumption that m II is intersected by a transversal is wrong.
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a polynomial finction with real coefficients has no real zeros. what can you say sbout the degree of the polynomial?explain
A polynomial function is a function in an equation, such as the quadratic equation, cubic equation, etc.
What is polynomial function?A polynomial function is a function in an equation, such as the quadratic equation, cubic equation, etc., that only uses non-negative integer powers or only positive integer exponents of a variable. A polynomial with an exponent of 1 is, for instance, 2x+5. A polynomial is a function that has the formula f(x) = anxn + anxn + anxn+1 +... + a2x2 + a1x + a0. The highest power of x in the formulation of a polynomial is its degree. Constant (non-zero) polynomials, linear polynomials, quadratics, cubics, and quartics, respectively, are polynomials of degree 0, 1, 2, 3, and 4.A polynomial is an expression in mathematics that solely uses the operations of addition, subtraction, multiplication, and powers of positive-integer variables. It consists of indeterminates and coefficients.
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if two events are 300s apart on clock a how far apart are they on a clock b and b clock c when clock a reads 380
The difference between two events that are 300s apart on clock A will also be 300s apart on clock B and clock C when clock A reads 380s. The time difference between events is independent of the reading on any individual clock.
The time on any of these clocks is a straight-line function of that on another, with slopes 1 and y-intercepts. Use the equation of the straight line to calculate the above quantities.
If two figures are similar, then the ratio of its corresponding sides is equal to the scale factor.
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MATH PROLBEM FOR 53 POINTS
If EC = 13, and AD = 10, find DC.
The length of DC is 24.
What is Pythagoras Theorem?Pythagoras theorem states that for a right angled triangle, the square of the hypotenuse is the sum of the squares of base and altitude.
Given is a figure of a quadrilateral.
Given that EC = 13.
Then AE = 13, since the diagonals bisect each other.
AC = 13 + 13 = 26
AD = 10
Using Pythagoras Theorem,
(AC)² = (AD)² + (DC)²
26² = 10² + (DC)²
(DC)² = 26² - 10²
= 576
DC = √576 = 24
Hence the length of the side DC is 24,.
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What’s the length side of AB in ABC
the answer to this question is 4,5,6
Please help me with this question.
E ∪ F and E' ∩ F' are not mutually exclusive events.
What is mutually exclusive events?If two things can't happen at the same moment, they are mutually exclusive. Disjoint is another word that denotes mutual exclusion. When two occurrences are unrelated, there is no chance that both will happen at the same moment.
Given:
S= {1, 2, 3}
and E= {1}, F= {2, 3}
We know that the union contain all elements of sets.
E ∪ F = {1, 2, 3} = S
and. E' = S - E
= {2, 3}
and, F'= S - F
= {1}
Now, intersection contain the common elements of both sets
Then, E' ∩ F' = {1, 2, 3}
As, E ∪ F = E' ∩ F' are not mutually exclusive events.
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5x - 4y = 9 elimination
Answer:
x = 4y/5 + 9/5
y = 5x/4 - 9/4
Step-by-step explanation:
5x - 4y = 9 ==> solve for x
5x = 4y + 9 ==> add 4y on both sides to isolate x
x = (4y + 9) / 5 ==> divide both sides by 5 to isolate x
x = 4y/5 + 9/5 ==> distribution property
5x - 4y = 9 ==> solve for y
5x = 4y + 9 ==> add 4y on both sides to make y positive
5x - 9 = 4y ==> subtract 9 on both sides to isolate y
y = (5x - 9) / 4 ==> divide both sides by 4 to isolate y
y = 5x/4 - 9/4 ==> distribution property
Question content area top
Part 1
A car moving at a constant speed passed a timing device at tequals
0.
After 9
seconds, the car has traveled 828
ft. Write a linear function rule to model the distance in feet d the car has traveled any number of seconds t after passing the timing device.
A linear function rule to model the distance in feet {d} the car has traveled any number of seconds {t} after passing the timing device is
d = 92t.
What are algebraic expressions?In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.Given is that a car moving at a constant speed passed a timing device at t = 0 s. After 9s, the car has traveled 828 ft.
We can write the linear function as -
y = mx
Now, we can write the slope as -
{m} = (828/9) = 92
So -
d = 92t
Therefore, a linear function rule to model the distance in feet {d} the car has traveled any number of seconds {t} after passing the timing device is
d = 92t.
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garnet has a container with 3 1/2 cups of flour of flour in it. She can make one dozen muffins with 2 1/2 cups of flour. How many dozens of muffins can garnet make with the flour she has?
The dozens of muffins that garnet can make with the flour she has is;
1²/₅
How to solve Fraction Word Problems?Let us begin by saying that one dozen is 12 and not a bakers dozen (13).
Thus, our solution will be arrived as by first seeing how many times 2¹/₂ can go into 3¹/₂ by dividing the two.
Let us first convert the mixed fractions to improper fractions to get;
2¹/₂ = 5/2
3¹/₂ = 7/2
Thus, the final step is;
7/2 ÷ 5/2
= 7/5
= 1²/₅
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of the 400 employes 5% of them are manegers
Answer:
20%
Step-by-step explanation:
To answer this question we have to 5% of 400, this will give us the number of managers!
1) 5% of 400
To find 5% of 400 we have to do 5 × 400...5 × 400 = 2000Then 2000 ÷ 100...2000 ÷ 100 = 20%This gives us the answer!
Have a lovely day :)
How do i Simplify to a single power of 4
Figure is rotated, 90° clockwise about the origin, followed by a reflection in the X axis, and then translated one unit up and three units to the left
The series of transformations of the figure will be rotation, reflection, and translation shown in the figure.
What is a transformation of a shape?A point, line, or mathematical figure can be converted in one of four ways, and each has an effect on the object's structure and/or position.
Rotation does not change the shape and size of the geometry. But changes the orientation of the geometry.
The translation does not change the shape and size of the geometry. But changes the location.
The reflection does not change the shape and size of the geometry. But flipped the image.
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Solve the following system of equations algebraically for all values of x, y, and z. show work
2x + 3y - 4x = -1
x - 2y + 5z = 3
- 4x + y + z = 16
Answer:
(x, y, z) = (-2, 5, 3)
Step-by-step explanation:
You want an algebraic solution to the system of equations ...
2x + 3y - 4x = -1x - 2y + 5z = 3-4x + y + z = 16SubstitutionThe second equation suggests an expression for x:
x = 2y -5z +3
Substituting this into the first and third equations, we have ...
2(2y -5z +3) +3y -4z = -1 ⇒ 7y -14z = -7 ⇒ y -2z = -1
-4(2y -5z +3) +y +z = 16 ⇒ -7y +21z = 28 ⇒ y -3z = -4
These equations are reduced to standard form by dividing by the leading coefficient.
EliminationSubtracting the second of these equations from the first, we have ...
(y -2z) -(y -3z) = (-1) -(-4)
z = 3 . . . . . . . simplify
The first of these reduced equations tells us ...
y = 2z -1 = 2(3) -1 = 5
And our original substitution equation says ...
x = 2y -5z +3 = 2(5) -5(3) +3 = -2
The solution is (x, y, z) = (-2, 5, 3).
__
Additional comment
A calculator confirms this solution. The "work" is entering the correct coefficients into the appropriate calculator function, then interpreting the result.
Find the surface area of the box shown..
Answer:
304 in²
Step-by-step explanation:
You want the surface area of a rectangular prism with dimensions 4 in, 8 in, and 10 in.
AreaThe area of a rectangular prism is given by the formula ...
A = 2(LW +H(L +W))
Using the given dimensions, the area is ...
A = 2(4·8 +10(4+8)) = 2(32 +120) = 304 . . . . square inches
The area of the box is 304 square inches.
How many letters do you need to name a point? a. One b. Two c. Three d. Four
The correct option is option(a) as only one letter is required to name a point.
What is a point?
A point in traditional Euclidean geometry is a simple concept that represents an exact place in space and has no length, breadth, or thickness. A point in modern mathematics is more broadly referred to as a component of a set known as a space. A point is an area that is depicted by a dot. A point just has a position; it lacks any dimensions such as length, width, shape, or size.
Usually, a geometrical figure is named by assigning one letter to each vertex of the figure.
So for a rectangle, it is 4 letters.
Hexagon is named using 6 letters since it has 6 vertices.
For a point, as mentioned there are no dimensions. It is just a dot.
Therefore a point is named using only one letter.
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The mortgage on your house is five years old. It required monthly payments of $1,402 , had an original term of 30 years, and had an interest rate of 10% (APR). In the intervening five years, interest rates have fallen and so you have decided to refinancethat is, you will roll over the outstanding balance into a new mortgage. The new mortgage has a 30-year term, requires monthly payments, and has an interest rate of 6.625% (APR).
Answer: Without more information about the outstanding balance on your mortgage and the exact terms of the new mortgage, it is not possible to determine the answer to your question. The amount of the monthly payments and the total amount of interest paid over the life of the loan will depend on these factors. But here a way you can do it.
Step-by-step explanation:
First, determine the outstanding balance on the original mortgage by calculating the total payments made over the past five years and subtracting them from the original loan amount. To do this, multiply the monthly payment amount by the number of months in five years (60) to find the total payments made.
Next, calculate the new monthly payment amount on the refinance mortgage by using a mortgage calculator or the formula: M = P[r(1+r)^n]/[(1+r)^n-1], where M is the monthly payment, P is the outstanding balance from step 1, r is the interest rate (0.06625), and n is the number of payments (360 for a 30-year mortgage).
Compare the new monthly payment amount to the original monthly payment amount to see how much you will save or pay more each month.
Consider any fees or closing costs associated with refinancing and factor those into the decision to refinance.
If the savings from the lower interest rate outweigh the costs of refinancing, it may be a good decision to move forward with the refinance.
select all the dxpressions equivalent to 7^-2 • 7^5 • 7^-3
The solution of the expression ( 7⁻² x 7⁵ x 7⁻³ ) is equal to 1.
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that the expression is ( 7⁻² x 7⁵ x 7⁻³ ). The expression will be solved as below,
E = ( 7⁻² x 7⁵ x 7⁻³ )
E = 7⁽ ⁻²⁺⁵⁻³⁾
E = 7⁽⁻⁵⁺⁵⁾
E = 7⁰
E =1
Hence, the solution of the expression ( 7⁻² x 7⁵ x 7⁻³ ) is equal to 1.
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Please help me with this question.
Answer:
sinB= 3.8(sin16/2.4) or ~0.45162
[Note: ~ means about]
Step-by-step explanation:
To solve this problem, we need to use the Law of Sines. The Law Of Sines states that sinA/a = sinB/b. Since we know the value of A,a, and b, we can plug our known values in to find our unknown value, sinB.
sinA/a = sinB/b [Plug in known values]
sin16/2.4 = sinB/3.8
Now, we want to just find the value of sinB. To do this, we will need to multiply both sides by 3.8, so the right side just reads sinB.
sin16/2.4 = sinB/3.8 [Multiply by 3.8]
3.8(sin16/2.4) = sinB
Now, if we want the exact answer, we can leave it as is. If you are looking for a decimal answer, let's continue to see what we would get:
3.8(sin16/2.4) = sinB
3.8(0.27563/2.4) ~ sinB
3.8(0.11484) ~ sinB
0.43642 ~ sinB
B ~ 0.45162
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Hi! Can someone please help with this question? I'm really confused on how exactly to find the force of magnitude
The magnitude and direction of the force C are 93.853 pounds-force and 281.237°, respectively.
How to find the magnitude and direction of a force in a 3-vector system
In this problem we find the case of a system at equilibrium and formed by 3 vectors, of which we must determine the magnitude and direction of a force must be determined. Vectors are described by two features: (i) Magnitude, (ii) Direction.
The magnitude is obtained by Pythagorean theorem:
F = √(x² + y²)
The direction is determined by inverse trigonometric functions:
θ = tan⁻¹ (y / x)
Besides, the system can be described physically by Newton's laws and mathematically by vector sum:
F₁(x, y) + F₂(x, y) + F₃(x, y) = (0, 0)
First, determine the vector associated to force C:
F₃(x, y) = - F₁(x, y) - F₂(x, y)
F₃(x, y) = - (- 92 · cos 45°, 92 · sin 45°) - (54 · cos 30°, 54 · sin 30°)
F₃(x, y) = - (- 46√2, 46√2) - (27√3, 27)
F₃(x, y) = (46√2, - 46√2) - (27√3, 27)
F₃(x, y) = (46√2 - 27√3, - 46√2 - 27)
Second, calculate the magnitude of force C:
F₃ = √[(46√2 - 27√3)² + (- 46√2 - 27)²]
F₃ ≈ 93.853
Third, determine the direction of force C:
θ = tan⁻¹ [(- 46√2 - 27) / (46√2 - 27√3)]
θ ≈ 281.237°
To learn more on Newton's laws: https://brainly.com/question/13678295
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Answer:
F₃ = 94 lb at 78.8° below the +x axis
Step-by-step explanation:
You want the magnitude of force F₃ that makes the sum of forces be zero.
Sum of forcesAs you have already noted, the sum of forces is zero when the system is at equilibrium.
F₁ +F₂ +F₃ = 0
F₃ = -(F₁ +F₂)
The forces F₁ and F₂ can be added by adding their horizontal and vertical components.
F₁ = 92(cos(135°), sin(135°)) ≈ (-65.0538, 65.0538)
F₂ = 54(cos(30°), sin(30°)) ≈ (46.7654, 27)
Then the sum of forces is ...
F₁ +F₂ ≈ (-18.2884, 92.0538) . . . . . sum of corresponding parts
MagnitudeAs we noted above, F₃ is the opposite of the sum, so is ...
F₃ = (18.2884, -92.0538)
The magnitude is found using the Pythagorean theorem:
|F₃| = √(18.2884² +(-92.0538)²) ≈ √8808.3677 ≈ 93.8529
The magnitude of F₃ is about 94 lb.
DirectionThe components of F₃ tell you the force is in the 4th quadrant. The ratio of the x- and y-components is the tangent of the angle:
θ = arctan(y/x) = arctan(-92.0538/18.2884) = arctan(-5.03345)
θ ≈ -78.763°
In the first attachment, the sum of forces F₁ +F₂ is shown by the black arrow. The direction of F₃ will be the opposite of that arrow's direction, so will be down to the right, about 78.8° below the +x axis. The second attachment shows the magnitude and direction of F₃, where 0° is the direction of the +x axis.
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Additional comment
For finding the magnitude of the sum of two forces, the Law of Cosines is another way to do it. To use that, you need to know the angle between the two forces. That's the angle at lower right inside the triangle shown in the first attachment. It is 75°. Then the magnitude is ...
|F₃| = √(54² +92² -2·54·92·cos(75°)) ≈ √8808.37 ≈ 93.853 . . . . lb
This math can be simpler than working with the x- and y-components of the forces.
The angle of F₃ is another matter. Finding that requires solving the force triangle for one of the other interior angles, then adding the angles in the appropriate way.
For example, the angle at the origin can be found from the Law of Sines as ...
α = arcsin(92/93.853·sin(75°)) ≈ 71.237°
Adding this to the angle of the red vector (30°) gives the resultant angle relative to the +x axis of 101.237°. The direction of F₃ is opposite this, so is ...
101.237° -180° = -78.763° . . . . as above
As you can see from the attachments, a vector calculator or spreadsheet is a useful tool for finding sums of forces from their magnitude and direction.
Just to be clear, the conversion between polar and rectangular forms is ...
(x, y) = r(cos(θ), sin(θ))
r = √(x² +y²)
θ = arctan(y/x) . . . . . with attention to quadrant
Spreadsheets and some calculators have an ATAN2(x, y) function that takes the signs into account when it computes the angle. When working with angles, make sure the calculator's angle mode is set to degrees or radians appropriately. (Spreadsheets usually use radians.)
