The formula to obtain the angle of rotation is as follows:
[tex]\cot 2\theta=\frac{A-C}{B}[/tex]Compare the given equation to the general equation of a conic.
[tex]Ax^2+Bxy+Cy^2+Dx+Ey+F=0[/tex]Thus, the values of A, B, and C are as follows.
[tex]\begin{gathered} A=32 \\ B=50 \\ C=7 \end{gathered}[/tex]Substitute the values into the equation.
[tex]\begin{gathered} \cot 2\theta=\frac{32-7}{50} \\ \cot 2\theta=\frac{25}{50} \\ \cot 2\theta=\frac{1}{2} \end{gathered}[/tex]Find the value of the θ.
[tex]\begin{gathered} \frac{1}{\tan 2\theta}=\frac{1}{2} \\ \tan 2\theta=2 \\ 2\theta=\tan ^{-1}(2) \\ 2\theta\approx63.4349 \\ \theta\approx31.7 \end{gathered}[/tex]Solve the following system of equations Detailed step by step
SOLUTION:
Step 1:
In this question, we are given the following:
[tex]\begin{gathered} 2\text{ x + y = 2 ------equation 1} \\ 4\text{ x + 3y =- 2--- -equation 2} \end{gathered}[/tex]Step 2:
The details of the solution are as follows:
The graphical solution for the two systems of equations are as follows:
CONCLUSION:
The solutions to the systems of equations are:
[tex]x\text{ = 4 , y = -6}[/tex]
2. Using Vièta's theorem, find the solutions to the equation. a) x^2 - 3x + 2 = 0 b) x^2 + 2x - 15 = 0.
Given:
[tex]\begin{gathered} x^2-3x+2=0 \\ x^2+2x-15=0 \end{gathered}[/tex]Required:
We need to find the solution by Vièta's theorem.
Explanation:
Compare 1st equation with
[tex]ax^2+bx+c=0[/tex]we get
[tex]\begin{gathered} a=1 \\ b=-3 \\ c=2 \end{gathered}[/tex]Vièta's theorem is
[tex]\begin{gathered} x_1+x_2=-\frac{b}{a} \\ x_1x_2=\frac{c}{a} \end{gathered}[/tex][tex]\begin{gathered} x_1+x_2=3 \\ x_1x_2=2 \end{gathered}[/tex]now solve this equation and we get
[tex]\begin{gathered} x_1=1 \\ x_2=2 \end{gathered}[/tex]because addition of 1 and 2 is 3 and multiplication is 2
Now for 2nd equation
[tex]\begin{gathered} a=1 \\ b=2 \\ c=-15 \end{gathered}[/tex]apply Vièta's theorem
[tex]\begin{gathered} x_1+x_2=-2 \\ x_1x_2=-15 \end{gathered}[/tex]by this
[tex]\begin{gathered} x_1=3 \\ x_2=-5 \end{gathered}[/tex]because addition of 3 and -5 is -2 and multiplication is -15
5. Prove triangle ABD is congruent to triangle CDB. DC|AB D
Enter the missing values in the area model to find 10(8y + 5)+510BoyAccording to the model above, 10(8y + 5) =Submit Answeatte
Note you have to use the value outside the bracket to multiply the inner value.
Hello! I need some help with this homework question, please? The question is posted in the image below. Q5
ANSWER:
STEP-BY-STEP EXPLANATION:
We can determine the domain and range of the function, knowing that the domain is the interval of values in x and the range is the interval of values in y.
Therefore:
[tex]\begin{gathered} D=\lbrack-5,5\rbrack \\ R=\mleft[-5,\frac{25}{17}\mright] \end{gathered}[/tex]When a function is inverted, the domain and range are inverted, therefore:
[tex]\begin{gathered} D=\mleft[-5,\frac{25}{17}\mright] \\ R=\lbrack-5,5\rbrack \end{gathered}[/tex]Which means that the inverted function goes from -5 to 25/17 in x and from -5 to 5 in y.
In addition to this we must take into account that when the inverse function is done, in most cases a reflection is made in y = x.
The only graph that meets all of the above is graph A
Solve for the unknown: 6(B+2) = 30
The unknown is B
[tex]6(B+2)=30[/tex][tex]\begin{gathered} 6B+12=30 \\ 6B+12-12=30-12 \\ 6B=18 \\ B=\frac{18}{6} \\ B=3 \end{gathered}[/tex]A student is trying to solve the set of two equations given below:Equation A: x + z = 6Equation B: 3x + 2z = 1Which of the following is a possible step used in eliminating the z-term
Answer:
multiply equation A by -2
Given these points please solve this problme.
The point that belongs to the solution set is A( 4, 4)
What are inequalities?Inequalities are defined as mathematical relations involving an unequal comparison between two numbers, elements or other arithmetic expressions.
They are mostly used to compare two numbers on the number line on the basis of their sizes.
Given the inequalities;
x + y > 63x - 5y ≤ 2Make 'x' the subject from equation 1, we have;
x > 6 - y
substitute the value into equation 2, we have;
3( 6 - y) - 5y ≤ 2
expand the bracket
18 - 3y - 5y ≤ 2
collect like terms
- 8y ≤ 2 - 18
- 8y ≤ -16
Make 'y' the subject of formula
y ≤ 2
Substitute the value in equation 3
x > 6 - 2
x > 4
Hence, the point is A( 4, 4)
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Professor Torres is stucked in a burning building. He is leaning to the window on the 5th floor which is 60feets above the ground. For stability, the firefighter have to place the bottom of their ladder 15feet from the wall of the building. How long does the ladder needs to be to reach the window on the 5th floor and save professor Torres? round to 2 decimal place
The situation forms the right triangle above:
Where x is the length of the ladder.
Apply the Pythagorean theorem:
c^2 = a^2 +b^1
where:
c = hypotenuse = longest side = x
A &b = the other 2 legs of the triangle
Replacing:
x^2 = 60^2 + 15^2
Solve for x
x^2 = 3,600 + 225
x^2 = 3,825
x =√3,825
x = 61.85 ft
3) Describe what ALL graphs of proportional relationships have in common
SOLUTION
What all graphs of proportional relationships have in common is a straight line.
This line is straight, no curves or bends. This straight line passes through the origin at an intersection of
[tex](0,0)[/tex]Hence, the answer is "A straight line that passes through the origin and goes at a constant rate".
name a 2 digit odd number that is composite
We should know that:
All the odd integers which are not prime are odd composite numbers. Examples of composite odd numbers are 9, 15, 21, 25
which graph represents the solution to -1/2m>7/11
The graph of the inequality:
(-1/2)*m > 7/11
Can be seen in the image at the end.
Which graph represents the solution for the inequality?Here we have the following inequality:
(-1/2)*m > 7/11
First, let's solve this for m, this means that we need to isolate the variable in one side of the inequality.
If we multiply both sides by -2, we get:
-2*(-1/2)*m < -2*(7/11)
Where the direction of the symbol changes because we are multiplying by a negative number.
m < -14/11
The graph of this will be an open circle at -14/11 and an arrow that goes to the left, like the one below.
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Identify the slope and y-intercept of the line y=−2x−1.
Given the equation of the line:
[tex]y=-2x-1[/tex]The given equation as the slope-intercept form: y = m * x + b
where m is the slope and b is the y-intercept
the y-intercept is the value of y when x = 0
By comparing the given equation with the slope-intercept form
So,
The slope = m = -2
The y-intercept = b = -1
the point of y-intercept = ( 0, -1 )
Help with these two questions please. Match the sentence with a word
EXPLANATION
Given that two angles form a linear pair, we can assevere that the postulate that applies is the Linear Pair Postulate.
Solve, graph and write the solution in interval notation: |2x−1|>5
Given: the inequality is,
[tex]|2x-1|>5[/tex]To solve the inequality,
[tex]\begin{gathered} |2x-1|>5 \\ -5<2x-1<5 \\ -5+1<2x<5+1 \\ -4<2x<6 \\ -\frac{4}{2}The graph will conntain a region -2The graph for the giev inequality is,
(4, 2); slope = 3 writing linear equations given point and slope
The standard equation of a line in point-slope form is expressed as;
y-y0 = m(x-x0) where;
m is the slope
(x0, y0) is the point on the line
Given
(x0, y0) = (4, 2)
From the coordinate;
x0 = 4 and y0 =2
slope m = 3
Substitute the given parameters into the equation as shown;
y-2 = 3(x-4)
Hence the linear equations given the point and slope id expressed as y-2 = 3(x-4)
Elana has 80 unit squares. What is the volume of the largest cube she can build with them? Need to show work to explain to my son, having a hard time with this.
Answer: The largest cube has volume of 64 cubic units, and the sides are 4 units long.
Step-by-step explanation:
Elena has 80 unit cubes and she has to build the largest cube using the unit cubes she has
Unit cube has a dimension of 1 unit on each side (Cube has all sides equal)
To make the largest cube, she needs to calculate the maximum volume which is near 80 units of cubes
Therefore,
We have a cube with each side 4 units whose volume is 64 and a cube with each side 5 units whose volume is 125
Elena has only 80 unit cubes to build the maximum-sized cube
Therefore she will be able to build a cube with each side as 4 units with a volume of 64 units with 16 spare cubes
A car is purchased for 19,00. Each year it loses 25% of its value. After how many years will the car be worth 5800. dollars or less? Write the smallest possible whole number answer
5 years
Explanation
Given
Cost price = $ 19,000
Depreciation yearly is % 25
What to find
Time to depreciate to $ 5, 800 or less
Step- by - Step Solution
After first year St
[tex]\begin{gathered} 25\%\text{ }of\text{ 19,000} \\ \\ \frac{25}{100}\times\text{ 19,000 = 4,750} \\ \\ 19,000\text{ - 4750 = 14, 250} \end{gathered}[/tex]After the year the second year
[tex]\begin{gathered} \frac{25}{100}\text{ }\times\text{ 14, 250 = 3,562.5} \\ \\ 14,250\text{ - 3,562.5 =10, 687.5} \end{gathered}[/tex]After Third year
[tex]\begin{gathered} 25\%\text{ of 10,687.5} \\ \\ \frac{25}{100\text{ }}\times\text{ 10, 687.5 = 2,671.875} \\ \\ 10,687.5\text{ - 2,671.875 = 8,015.625} \\ \end{gathered}[/tex]After Fourth year
[tex]\begin{gathered} 25\%\text{ of 8,015.625} \\ \\ \frac{25}{100}\times\text{ 8,015.625 = 20003.906} \\ \\ 8\text{,015.625 - 20,003.906 = 6011.719} \end{gathered}[/tex]After Fifth year
[tex]\begin{gathered} 25\%\text{ of 6011.719} \\ \\ \frac{25}{100}\times\text{ 6011.719 = 1502.930} \\ \\ 6011.719-1502.930\text{ = 4508.789} \end{gathered}[/tex]Therefore after 5 years the car be worth 5800. dollars or less Therefore after 5 years the car be worth 5800. dollars or less
An exam has 2 papers each scored differently. one is out of 120 and another is out of 80. Maryam scores 65% on the first and 80% on the second. work Maryam's total percentage score for her exam.
Maryam's total percentage score on her exam is 71%.
What is the total percentage score?
Percentage is the ratio of an amount that is expressed as a number out of hundred. The sign that is used to represent percentage is %.
The first step is to determine the score on each paper.
Score on the first test = 65% x 120
(65 / 100) x 120 = 78
Score on the second test = 80% x 80
0.80 x 80 = 64
Total percentage score = (sum of scores / total score) x 100
Sum of scores = 64 + 78 = 142
Total score = 120 + 80 = 200
(142 / 200) x 100 = 71%
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-8(4 p -1)-7 p+8( p+1)
-31p + 16
Explanation:
-8(4 p -1) - 7p+8( p+1)
Open the bracket:
-8(4p) -8(-1) - 7p +8(p) +8(+1)
Simplify:
-32p + 8 - 7p + 8p + 8
Note: the multiplication of opposite sign gives negative number. While multiplication of same sign gives positive number
Collect like terms:
= -32p - 7p + 8p + 8 + 8
= -31p + 16
what is the area of a circle with the radius of 10, then rounding the answer to the nearest tenth?
Given a circle with a radius = r = 10 in
The area of the circle is given by the following formula:
[tex]A=\pi\cdot r^2[/tex]Substitute with r= 10
so, the area will be:
[tex]A=\pi\cdot10^2=\frac{22}{7}\cdot10^2=\frac{22}{7}\cdot100=314.2857[/tex]Rounding the answer to the nearest tenth:
So, the answer will be area = 314.3 square inches
In a group of 80 animals, 32 are dogs. Dogs make upwhat percent of the animals in the group?A. 32.00B. 28.6C. 35.5D. 38.00E 40.00
Let's calculate the percentage of dogs in the animal group
[tex]\begin{gathered} P=\frac{32}{80} \\ P=0.40 \\ P=40\text{ \%} \end{gathered}[/tex]The answer would be 40%.
The equation 3x + 2y = 120 models the number of passengers who can sit in a train car, where isthe number of adults and y is the number of children. Explain what the 2- and y-intercepts mean.
Explanation:
Given the equation that models the number of passengers who can sit in a car expressed as 3x + 2y = 120
x is the number of adults
y is the number of children
The x-intercept is the point where y is zero i.e. the number of adults when there is no number of children.
when y = 0
3x + 2(0) = 120
3x = 120
x = 120/3
x = 40
This means that there will be 40 adults if there are no children
The y-intercept is the point where x is zero i.e. the number of children when there is no number of adults.
when x = 0
3(0) + 2y = 120
2y = 120
y = 120/2
y = 60
This means that there will be 60children if there are no adults
2064 is divisible by 2, 4 and 8 true or false
describe the center and spread of the data using the more appropriate status either the mean median range interquartile range or standard division
My Marjorie made for rates and 6 hours and 6 wreaths and 9 hours what is the constant of proportionality
The constant of proportionality is computed as follows:
[tex]k=\frac{\text{number of wreaths}}{\text{ number of hours}}[/tex]Assuming that 6 wreaths correspond to 9 hours, the constant of proportionality is:
[tex]k=\frac{6\text{ wreaths}}{9\text{ hours}}=\frac{2}{3}\frac{wreath}{hour}[/tex]Find y if the line through (1, y) and (8, 2) has a slope of 3.
[tex](\stackrel{x_1}{1}~,~\stackrel{y_1}{y})\qquad (\stackrel{x_2}{8}~,~\stackrel{y_2}{2}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{2}-\stackrel{y1}{y}}}{\underset{run} {\underset{x_2}{8}-\underset{x_1}{1}}} ~~ = ~~\stackrel{\stackrel{m}{\downarrow }}{3}\implies \cfrac{2-y}{7}=3 \\\\\\ 2-y=21\implies -y=19\implies y=\cfrac{19}{-1}\implies y=-19[/tex]
estimate 2,829 divided by 33=?
Answer: 100
Step-by-step explanation:
Calculate it and 85.7272727273 is closer to 100 so its 100
The area of a circle is about 167.3306 square inches. The circle's circumference is ____ inches.Use 3.14 for π.
The area of a circle can be calculated using this formula:
[tex]A=\pi r^2[/tex]Where "r" is the radius of the circle.
The circumference of a circle can be found using this formula:
[tex]C=2\pi r[/tex]Where "r" is the radius of the circle.
In this case you know that the area of this circle is:
[tex]A\approx167.3306in^2[/tex]Then, you can substitute this value into the first formula and solve for "r". Use:
[tex]\pi=3.14[/tex]Then:
[tex]\begin{gathered} (167.3306in^2)=(3.14)r^2 \\ \\ \frac{(167.3306in^2)}{3.14}=r^2 \\ \\ r=\sqrt[]{(\frac{167.3306in^2}{3.14})} \\ \\ r=7.3in \end{gathered}[/tex]Now you can substitute this value into the formula for calculate the circumference of a circle:
[tex]\begin{gathered} C=(2)(3.14)(7.3in) \\ \end{gathered}[/tex]Finally, evaluating, you get:
[tex]C=45.844in[/tex]The answer is:
[tex]45.844in[/tex]the net of a rectangular prism is shown below. the surface area of each face is labeled. which vakues represent the dimensions, in meters, of the rectangular prism.
The answer is 5, 9, 10