In this problem, you have a square with side length 6 and four smaller squares with side length 2 placed in the corners. The points W, X, Y, and Z are the vertices of the smaller squares. The goal is to find the maximum possible distance between points A and P.
[asy] size(200); pair A,B,C,D,P,W,X,Y,Z; A=(0,4); B=(4,4); C=(4,0); D=(0,0); P=(2,2); W=(0,6); X=(6,6); Y=(6,0); Z=(0,0); draw(A--B--C--D--cycle); draw(W--X--Y--Z--cycle); draw(A--P--C); label("A",A,NW); label("B",B,NE); label("C",C,SE); label("D",D,SW); label("P",P,S); label("W",W,N); label("X",X,NE); label("Y",Y,S); label("Z",Z,SW); [/asy]
The point A is located on the bottom-left corner of the square with side length 6, and the point P is the center of the square $ABCD$. To find the maximum distance from A to P, we need to find the longest diagonal of the square $ABCD$.
The longest diagonal of a square is the one that goes from one corner of the square to the opposite corner. In this case, the longest diagonal goes from point A to point C, which is a distance of 6 units. Therefore, the maximum possible distance from A to P is 6 units.
The following dot plot shows the pulse rates of runners after finishing a marathonWhich of the following data sets is represented in the dot plot?A. {145, 165, 175, 1853}B. {145, 150, 150, 152, 153, 160, 163, 165, 170, 170, 178, 179, 185}C. {1, 5, 6, 1}D. {145, 165, 165, 165, 165, 165, 175, 175, 175, 175, 175, 175, 185}
The data set that represented by the dot plot given is:
D. {145, 165, 165, 165, 165, 165, 175, 175, 175, 175, 175, 175, 185}
In a dot plot, each data point in a data set is represented by a dot.
To find out the data set that is represented in the dot plot, list out the data points as plotted on the dot plot given.
In the given dot plot that is showing the pulse rates of runners after finishing a marathon, it can be seen that,
There are 1 dot on 145, 5 dots on 165, 6 dots on 175 and 1 dot on 185.
Thus:
145 - 1 dot = 145165 - 5 dots = 165, 165, 165, 165, 165175 - 6 dots = 175, 175, 175, 175, 175, 175185 - 1 dot = 185Therefore, the data set that represented by the dot plot given is:
D. {145, 165, 165, 165, 165, 165, 175, 175, 175, 175, 175, 175, 185}
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write the missing angle measure next to each letter.
please help girl
The measure of the missing angles are a = 65°, b = 115°, c = 115°, d = 65° and e = 65°
How to write the missing angle measure?Angle geometry is a branch of mathematics that deals with angles and the properties of angles in geometric shapes such as measuring angles, classifying angles, and solving problems involving angles.
The measure of the missing angles are as follow:
a = 65° (corresponding angles are equal)
a + b = 180° (sum of angles of straight line is 180°)
65° + b = 180°
b = 180-65 = 115°
c = b (corresponding angles are equal)
c = 115°
d = 180 - c (sum of angles of straight line is 180°)
d = 180 - 115 = 65°
e = d (corresponding angles are equal)
e = 65°
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pls help I'm failing :(
Kasey tries to run a certain number of miles by the end of the month. if kasey is 60% close to reaching her goal and has already run 24 miles, how many miles is she trying to run by the end of the month?
Kasey is trying to run 60 miles by the end of the month.
Let us assume that Kasey tries to run 't' miles by the end of the month.
Kasey is 60% close to reaching her goal.
This means, she has already completed 40% of her goal.
Foven that she has already run 24 miles.
This means 40 percent of total distance is 24 miles.
i.e., 40 percent of 't' miles = 24 miles
We calculate 40 percent of t.
(40/100) × t = 24
(2/5) × t = 24
2 × t = 24 × 5
2 × t = 120
t = 120/2
t = 60 miles
i.e., she tries to run a 60 miles by the end of the month.
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Find the Perimeter of a box with the area of 96 square feet?
Answer:
24
Step-by-step explanation:
There are 4 sides to a box. Meaning you will have to divide 96 by 4. Giving you 24. Since perimeter means to add, add 24 4 times and you will be given the result of 96.
A circle is circumscribed around $ABCD$ as follows:How many degrees are in $\angle CAB \angle ACD$
The measure of [tex]$\angle CAB = 180^{\circ} - \angle ACD$[/tex]
What is the circumscribed circle?
A circumscribed circle is a circle that is drawn around a polygon such that the circle touches all the vertices of the polygon. In other words, the vertices of the polygon lie on the circumference of the circle.
Given that a circle is circumscribed around a quadrilateral ABCD, it is clear that the circle passes through all four vertices of the quadrilateral, A, B, C and, D. Also, the diameter of the circle is the diagonal AC of the quadrilateral.
The angles formed by the diameter with the tangents from the non-adjacent vertices are supplementary. That is [tex]$\angle CAB + \angle ACD = 180^{\circ}$[/tex]
Therefore, the measure of [tex]$\angle CAB = 180^{\circ} - \angle ACD$[/tex]
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Rewrite the expression with a rational exponent as a radical expression.
[tex]( {4}^{ \frac{2}{5} } {)}^{ \frac{1}{4} } [/tex]
The expression [tex](4^{(2/5)})^{(1/4)}[/tex] can be rewritten as the radical expression [tex]4^{(1/1280)}.[/tex]
What is the rational exponent?
A rational exponent is an exponent that is a fraction. In mathematics, a number raised to a fractional power represents a root of the number. For example, the expression[tex]x^{(1/3)}[/tex] represents the cube root of x. Similarly, the expression [tex]x^{(2/5)}[/tex] represents the fifth root of x².
The given expression can be rewritten as a radical expression as follows:
[tex](4^{(2/5)})^{(1/4)} = 4^{(2/5) * (1/4)}[/tex]
Using the property of exponents that (a^b)^c = a^(b*c), we simplify the right side of the equation:
[tex]4^{((2/5) * (1/4))} = 4^{(2/5 * 1/256)} = 4^{(1/1280)}[/tex]
So, the expression [tex](4^{(2/5)})^{(1/4)}[/tex] can be rewritten as the radical expression [tex]4^{(1/1280)}.[/tex]
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Nolan and his children went into a grocery store and he bought $9 worth of apples and bananas. Each apple costs $0.75 and each banana costs $0.50. He bought 2 more apples than bananas. By following the steps below, determine the number of apples, x, and the number of bananas, y, that Nolan bought.
Determine three ways to have 2 more apples than bananas:
Nolan and his children went into a grocery store and he bought $9 worth of apples and bananas, The three ways are explained below
What are equations?Generally, One way to determine the number of apples, x, and the number of bananas, y, that Nolan bought is to use the equation:
x + y = a (total number of apples and bananas)
0.75x + 0.5y = 9 (total cost of apples and bananas)
Another way is to use the equation:
x = y + 2 (number of apples is 2 more than the number of bananas) 0.75x + 0.5y = 9 (total cost of apples and bananas)
A third way is to use the equation
y = x - 2 (number of bananas is 2 less than the number of apples)
0.75x + 0.5(x-2) = 9 (total cost of apples and bananas)
You can then solve for x and y using either of the above equations.
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Part D & E answer please :)
D. The retail price of the phone is given as follows: 42.
E. The percent reduction in the price to $31.50 is given as follows: 25%.
How to obtain the prices and percentages?The prices and the percentages are obtained applying the proportions in the context of this problem.
The purchase price is given as follows:
$12.
The markup rate is given as follows:
250%.
Meaning that the retail price is of 100% + 250% = 350% of the purchase price, hence it's value is obtained as follows:
3.5 x 12 = $42.
The percentage reduction is given by the reduction in the price divided by the initial price and then multiplied by 100%.
The reduction from $42 to $31.50 is given as follows:
42 - 31.50 = $10.50.
Hence the percent reduction is given as follows:
10.5/42 x 100% = 25%.
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(a) On March 2, Marin Company sold $914,000 of merchandise to Cullumber Company on account, terms 3/10, n/30. The cost of the merchandise sold was $592,400. (b) On March 6, Cullumber Company returned $104,600 of the merchandise purchased on March 2. The cost of the merchandise returned was $64,400. (c) On March 12, Marin Company received the balance due from Cullumber Company.
The final answers are as follows gross profit $321,600, cost of goods sold of $528,000 and balance due $782,980.
(a) The terms of the sale are 3/10, n/30 which means that if Cullumber Company pays the invoice within 10 days, they will receive a 3% discount on the total invoice amount. However, if they pay within 30 days, they will not receive a discount.
To calculate the amount due if Cullumber Company takes the discount:
$914,000 x 3/100 = $27,420
$914,000 - $27,420 = $886,580
To calculate the gross profit:
$914,000 - $592,400 = $321,600
(b) The return of merchandise means that Cullumber Company is returning $104,600 worth of merchandise and thus the amount due to Marin will be:
$886,580 - $104,600 = $782,980
And the cost of goods sold will be:
$592,400 - $64,400 = $528,000
(c) On March 12, Marin Company received the balance due from Cullumber Company which is $782,980.
This means that the net amount received by Marin Company after the discount and return of merchandise is $782,980.
Therefore, The final answers are as follows gross profit $321,600, cost of goods sold of $528,000 and balance due $782,980.
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ms. young, a special education teacher, is teaching kim, a 13-year-old with multiple disabilities, how to identify each us coin (penny, nickel, dime, quarter). every day, ms. young gives kim 15 random coins to identify. with the help of a chart showing each coin and its name, kim is asked to identify each coin. the results after one week of instruction are shown below. day number of coins identified correctly monday 7 tuesday 4 wednesday 5 thursday 4 friday 2 based on this data, what should ms. young do next?
Ms. Young should use the data to decide on her next steps.
What is data in math?Data in math is any information that is used to make decisions and/or can be used to analyze patterns. It can be numerical, such as statistics, or qualitative, such as observational data. Data can be collected in various ways, including surveys, experiments, and observations. Data can also be represented graphically, such as in charts or diagrams, to help better illustrate the patterns or trends it may contain. Data is often used in the context of problem solving and decision making and is a key element in the scientific method.
She should identify the areas of difficulty for Kim, where she is having trouble identifying coins. She should then design lessons that focus on these areas, providing more practice on identifying coins. Using a variety of instructional strategies, such as hands-on activities, games, and visual aids, can help Kim to better understand the differences between the coins and increase her accuracy. Additionally, Ms. Young should provide more frequent feedback to Kim on her performance to ensure that she is making progress. Finally, Ms. Young should consider decreasing the number of coins presented to Kim each day, to give her more time to practice and increase her accuracy.
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Are lines L1 and L2 parallel? Are lines L1 and L4 perpendicular? Explain your thinking.
Answer:
yes dont listen to anything isay because i just wants the points its to harda for me
Step-by-step explanation:
A park has t tables spread equally among the 3 picnic areas.Write an expression that shows how many picnic tables are at each picnic area.
Answer:
8000
Step-by-step explanation:
bc if its 3 picnic its its 8000
Question 1 (Essay Worth 10 points)
(02.08 HC)
0
In a volatile housing market, the overall value of a home can be modeled by V(x) = 325x² - 4600x+145000, where V represents the value of the home
and x represents each year after 2020. Find the vertex and interpret what the vertex of this function means in terms of the value of the home. Show the
work you completed to determine the vertex.
Answer:
The vertex of a parabolic function like V(x) = 325x² - 4600x + 145000 can be found by using the vertex form of the equation, which is y = a(x-h)² + k. In this form, (h,k) is the vertex of the parabola. To convert the equation V(x) = 325x² - 4600x + 145000 into vertex form, we can complete the square to get the following:
V(x) = 325(x² - 14x) + 145000
To complete the square we will add and subtract (14/2)² = 49
V(x) = 325(x-7)² + 145000 - 49*325 = 325(x-7)² + 138375
So the vertex of the parabola is (h,k) = (7, 138375).
The vertex of a parabolic function represents the minimum or maximum point of the parabola, depending on whether the parabola opens upwards or downwards. In this case, the vertex represents the maximum value of the home. It means that the value of the home reaches its highest point at 7 years after 2020, and this highest point is 138375.
So in terms of the value of the home, the vertex of the function represents the point at which the housing market reaches its peak, after which the value of the home begins to decrease.
Pls help I need it Right now
The linear regression equation is given as follows:
y = -30x + 1048.8.
The estimate for the number of cases in 2026 is given as follows:
689 new cases.
How to obtain the equation for the regression line?The equation for the line of fit is obtained inserting the points of the data-set into a linear regression calculator.
(this is because there are multiple types of regression, such as logistic, exponential and quadratic regression).
The points for this problem are given as follows:
(0, 1047), (1, 1020), (2, 989), (3,943), (4,964), (5,880).
Inserting these points into a calculator, the regression equation is given as follows:
y = -30x + 1048.8.
2026 is 12 years after 2014, hence the estimate is calculated as follows:
y = -30(12) + 1048.8
y = 689 new cases.
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Using banker's algorithm what is safe sequence.(A) P2, P3, P1 (B) P1, P2, P3 (C) P3, P2, P1 (D) None of the above
P2, P3, P1 is a safe sequence according to the banker's algorithm.
What is banker's algorithm?The banker's algorithm gets its name from the fact that it determines whether or not a person should be granted a loan amount in order to assist the bank system in securely simulating resource allocation. In this part, we'll go through the Banker's Algorithm in depth. The Banker's Algorithm is primarily used in the banking system to avoid stalemate. It assists you in determining whether or not a loan will be granted. This procedure is used to assess the allocation for securely replicating the maximum amount available for all resources.
Here,
According to the banker's algorithm, the sequence P2, P3, P1 is secure.
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each half of this figure is composed of 3 red triangles, 5 blue triangles and 8 white triangles. when the upper half is folded down over the centerline, 2 pairs of red triangles coincide, as do 3 pairs of blue triangles. there are 2 red-white pairs. how many white pairs coincide?
7 white pairs coincide when upper half is folded over the centerline.
What is triangle ?
A triangle is a three-sided polygon that has three straight sides and three angles. The sum of the angles in a triangle is always 180 degrees. Triangles are classified based on their side lengths or angle measures.
We know that each half of the figure is composed of 3 red triangles, 5 blue triangles and 8 white triangles.
When the upper half is folded down over the centerline, 2 pairs of red triangles coincide, as do 3 pairs of blue triangles. there are 2 red-white pairs.
So, 2 red triangles + 3 blue triangles + 2 red-white pairs = 7 pairs of triangles that coincide
The total number of triangles in each half of the figure is 3 red triangles + 5 blue triangles + 8 white triangles = 16
If 7 pairs of triangles coincide, then 16 - 7 = 9 pairs do not coincide.
And since 2 red-white pairs coincide, then the remaining 9 - 2 = 7 pairs must be white pairs.
So 7 white pairs coincide when upper half is folded over the centerline.
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Zack deposited $4,533 in an account earning 5.9% interest compounded annually.
To the nearest cent, how much will he have in 1 year?
If Zack deposited $4,533 in an account earning 5.9% interest compounded annually. The amount he will have in 1 year is: $4,807.79.
How to find the compounded amount?Using this formula to find the compounded amount
A = P(1+r/n)^nt
Where:
A = Amount = ?
P = Principal = $4,533
R = Interest rate = 5.9% or 0.059
n = Number of months = 12
t = Time = 1 years
Let plug in the formula
A = $4,533 (1+ 0.059/12)^1×12
A = $4,533 (1+.004916)^12
A = $4,533 (1.004916)^12
A = $4,533 (1.0606)
A = $4,807.79
Therefore the amount he will have in 1 year is: $4,807.79.
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Are the ratios 1/2 and 2/3 equal?
No these ratios 1/2 is not equal to 2/3 because 2/3 is not the multiple of 1/2
What is an equivalent ratios?
A ratio compares two quantities named as antecedent and consequent, by the means of division. For example, when we cook food, then each ingredient has to be added in a ratio. Thus, we can say, a ratio is used to express one quantity as a fraction of another quantity.
Two ratios are equivalent to each other if one of them can be expressed as the multiple of the other. Hence, to get the equivalent ratio of another ratio, we have to multiply the two quantities (antecedent and consequent) by the same number.
Given ratios are 2:1 and 3:1
we can write it as 2/1 and 3/1.
The lcm of 2 and 3 is 6.
Multiply denominator of both ratio with 6, we get
2/6 and 3/6 And we can see both the ratios are not equal.
Hence, there are not equivalent ratios.
4:1 and 8:3 are equivalent ratios -----> is false, because 4:1 is equivalent to 8:2
11:2 and 2:11 are equivalent ratios------> is false (because, are reciprocal ratios, not equivalent ratios)
3:1 and 9:3 are equivalent ratios ------> is true
because 3:1 multiply both sides by 3 -----> 3*3:1*3=9:3
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Which ordered pairs make the equation true?
4x−3y=−10
Select each correct answer.
(3, −1)
(−4, −2)
(1, 4)
(−1, 2)
Answer:
(- 4, - 2 ) and (- 1, 2 )
Step-by-step explanation:
to determine which ordered pairs make the equation true
substitute the x and y coordinates of each point into the left side of the equation and if equal to the right side then it is a solution.
(3, - 1 )
4(3) - 3(- 1) = 12 + 3 = 15 ≠ - 10 ← not true
(- 4, - 2 )
4(- 4) - 3(- 2) = - 16 + 6 = - 10 ← true
(1, 4 )
4(1) - 3(4) = 4 - 12 = - 8 ≠ - 10 ← not true
(- 1, 2 )
4(- 1) - 3(2) = - 4 - 6 = - 10 ← true
Circles with centers at $O$ and $P$ have radii 2 and 4, respectively, and are externally tangent. Points $A$ and $B$ on the circle with center $O$ and points $C$ and $D$ on the circle with center $P$ are such that $\overline{AD}$ and $\overline{BC}$ are common external tangents to the circles. What is the area of the concave hexagon $AOBCPD$
Therefore , the solution of the given problem of surface area comes out to be Concave hexagonal area equals 24.
Surface area definitionIts surface area serves as a proxy for how much overall space it occupies. The whole environment of a three-dimensional shape is taken into account when calculating its surface area. The overall size of something is its surface area. The volume of water in a cuboid can be determined by summing the faces on all of its six rectangular sides. To determine the box's measurements, apply the following formula: For 2lh, 2lw, but also 2hw, the surface is the identical (SA). The region is represented by the surface area of the muti form.
Here,
Given :
The Pythagorean theorem can be used to calculate the distance between it circle centers and the difference in the radii of two circles (4-2=2) to determine the lengths of the rectangle's sides.
We can calculate the area of rectangular as well as the two triangles using that. The regions of the these three forms are added to create the hexagon's surface area.
Let d be the distance between both the circle centers in the scenario described.
d² = (4+2)^2
d = √((4+2)^2) = √(36) = 6
Rectangle area = 26 divided by 12
Each triangle's area is equal to (1/2)(2d)/(1/2)(2*6), or 6
Hexagonal area equals 12 + 2 * 6 = 24.
Concave hexagonal area equals 24.
Therefore , the solution of the given problem of surface area comes out to be Concave hexagonal area equals 24.
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What is 2x+4y-2x simplified?
work out the value of v in the equation below . 9.2/v = tan 19 give your answer to 1d.p
Answer: 26.7
Step-by-step explanation:
[tex]\frac{9.2}{v}=\tan 19^{\circ}\\\\\frac{v}{9.2}=\frac{1}{\tan 19^{\circ}}\\\\v=\frac{9.2}{\tan 19^{\circ}}\\\\v \approx 26.7[/tex]
Answer: The given equation is 9.2/v = tan(19). To solve for v, we can first multiply both sides of the equation by v to get rid of the denominator on the left side. This gives us 9.2 = v * tan(19).
Next, we can divide 9.2 by tan(19) to find the value of v. The value of tan(19) is approximately 0.3420201, so 9.2 / 0.3420201 = 26.884956 (rounded to 1 decimal place).
So the value of v in the equation is approximately 26.9 (to 1 decimal place).
Step-by-step explanation:
A basket contain 3 red balls, 5 blue balls and 2 green balls. 2 ball are picked one after the other without replacement find the probability that: A. Both are red first is blue, the other is green Both are of different colour They are of the same color
Answer:
Step-by-step explanation:
Total Number of balls = 3+5+2=10
a)Probability both are red: 3/10 * 2/9 = 1/15
b)First ball blue, other green : 5/10 * 2/9 = 1/9
c) Both are different color : (3/10*7/9)+(5/10*5/9)+(2/10*8/9)
=21/90+25/90+16/90=62/90=31/45
d)Both are same color: (3/10*2/9)+(5/10*4/9)+(2/10*1/9)
=6/90+20/90+2/90 = 28/90 = 14/45
Someone help me, I don't understand this
The diameter of the DVD in centimeters found by adding together the ridges and valleys across the diameter is about 11.998 centimeters.
What is the diameter of a circle?The diameter of a circle is the measured distance between the outer rim of the circumference of the circle, passing through the center of the circle.
The number and dimensions of the ridges (bumps) and valleys (without bumps) on the DVD disc are as follows;
The number of bumps across the diameter = 73,000
The thickness of a bump = 0.00032 cm
The number of valleys across the diameter = 73,000
The thickness of valley = 0.000074 cm
The diameter of the circle left blank at the middle of the DVD disc = 4.26 cm
The diameter of the DVD, D, is therefore obtained by multiplying thickness of the bumps and valleys then adding the result to the diameter of the circle in the middle as follows;
D = 73,000 × 0.000032 cm + 73,000 × 0.000074 cm + 4.26 cm = 11.998 cm
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How many faces does this polyhedron have?
HELP!!
6
12
8
10
Answer:
6
Step-by-step explanation:
you have to count the 2d faces
Answer:
6 faces
Step-by-step explanation:
DCGH
BCDA
BCGF
ADHE
BFEA
FGHE
for some experiment, rats were put in a maze and time of completion was noted. the completion time was normally distributed with a mean of 1.5 and a standard deviation of 0.35. if 5 rats are selected, what is the probability that their total completion time in the maze for all rats is between 6 and 8 minutes? (hint: the mean and standard deviation are for one rat).
With a mean of 1.5 and a standard deviation of 0.35. if 5 rats are selected, the probability that their total completion time in the maze for all rats is between 6 and 8 minutes is 0.4455.
First, we need to convert the given mean and standard deviation from "one rat" to "five rats". The mean for five rats would be 5 x 1.5 = 7.5, and the standard deviation for five rats would be sqrt(5) x 0.35 = 0.87. Next, we need to convert the given time range of 6-8 minutes to the standard normal distribution, using the z-score formula: z = (x - mean) / standard deviation.
For 6 minutes, z = (6 - 7.5) / 0.87 = -1.72 ,For 8 minutes, z = (8 - 7.5) / 0.87 = 0.58 Finally, we can use a standard normal distribution table to find the probability that the rats will complete the maze in between 6 and 8 minutes. This is the area under the standard normal curve between -1.72 and 0.58. The probability is 0.4455.
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Pls answer soon !!!!!!!!!
The required solution of the given simultaneous equation is x = 5 and y = -9.
What are simultaneous linear equations?Simultaneous linear equations are two- or three-variable linear equations that can be solved together to arrive at a common solution.
here,
Given a system of equations,
6x + 5y = -15 - - - - -(1)
-4x - y = -11 - - - - (2)
Multiply equation 1 by 1 and multiply equaiton 2 by 5 then add both the equation,
6x + 5y -20x -5y = -15 -55
-14x = -70
x = 5
Now,
-4(5) - y = -11
-20 - y = -11
-y = -11 +20
y = -9
Thus, the required solution of the given simultaneous equation is x = 5 and y = -9.
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#3
The price of a watch was decreased by 35% and
sold for $75. 99. What was the original price of the
watch in dollars?
A. $26. 60
C. $116. 91
B. $49. 39
D. $217. 11
As per the unitary method, the original price of the watch is 217.11
The term unitary method is defined as a basic concept to find the value of a single unit and then using that value to find the value of multiple units.
Here we have given that the price of a watch was decreased by 35% and sold for $75. 99.
Let us consider that x be the original price of watch.
Here we have given that the amount of discount is calculated as,
=> 35% of x = 75.99
=> 35/100 of x = 75.99
When we simplify this then we get,
=> 0.35 of x = 75.99
Then the value of x is calculated as,
=> x = 75.99/0.35
=> x = 217.11
Therefore, the correct option is (D).
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M110) solve
csc ( to power 5 ) theta - 4 csc theta + 1 = 0
Answer:
The equation csc^5 (theta) - 4 csc (theta) + 1 = 0 can be solved by factoring the left-hand side of the equation.
csc^5 (theta) - 4 csc (theta) + 1 = 0
csc (theta) (csc^4 (theta) - 4) + 1 = 0
csc (theta) (csc^2 (theta) - 2)(csc^2 (theta) + 2) + 1 = 0
We can see that the last expression is a polynomial equation of the form (ax+1)(bx^2+c*x+d)=0 where x= csc(theta) , a=1, b=1, c= -2, d=2
Solving for x = 0, we get x=0 which is not a valid solution because csc(theta) is defined only for theta != k*pi where k is an integer.
Solving for x= -1/2 and x= 1/2, we get csc(theta) = -1/2 and csc(theta) = 1/2
To find theta, we can use the reciprocal cosecant function (csc) which is defined as 1/sin(theta)
csc(theta) = -1/2 => sin(theta) = -2
csc(theta) = 1/2 => sin(theta) = 2
Answer: 0 is cos^4(theta) = (4 - 1/sin(theta))(1-sin^2(theta))^2.
Step-by-step explanation: To solve the equation csc^5(theta) - 4csc(theta) + 1 = 0, we can use the identity csc^n(theta) = 1/sin^n(theta).
First, we can apply the identity:
1/sin^5(theta) - 4/sin(theta) + 1 = 0
Next, we can factor out sin(theta) from the first two terms:
sin(theta)(1/sin^4(theta) - 4) + 1 = 0
We can then use the identity sin^2(theta) = 1 - cos^2(theta) to rewrite the first term:
sin(theta)(1 - cos^4(theta)/(1-cos^2(theta))^2 - 4) + 1 = 0
We can then divide both sides of the equation by sin(theta), and we get:
1 - cos^4(theta)/(1-cos^2(theta))^2 - 4 + 1/sin(theta) = 0
We can then simplify this equation to get:
cos^4(theta)/(1-cos^2(theta))^2 = 4 - 1/sin(theta)
By using the identity cos^2(theta) = 1-sin^2(theta) we can simplify the equation further:
cos^4(theta) = (4 - 1/sin(theta))(1-sin^2(theta))^2
So to find the solution we can use double angle identities to simplify the right side of the equation.
Therefore, the solution to the equation csc^5(theta) - 4csc(theta) + 1 = 0 is cos^4(theta) = (4 - 1/sin(theta))(1-sin^2(theta))^2.
