Answer:
C.
Explanation:
Let's analyze each answer option:
Option A.
Taking into account the table, we can say that 45 boys play video games daily and 12 girls play video games daily, so more boys play video games daily than girls because 45 is greater than 12.
Option B.
In the same way, there is a total of 57 students that play video games daily and 43 that don't. Since 57 is a little higher than 50 (half of the 100 students surveyed), we can say that a little more than a half of the students surveyed play video games daily.
Option C.
There are 50 boys surveyed and from them 5 do not play video games daily, so the percentage of boys that don't play video games daily is:
[tex]\frac{5}{50}\times100=0.1\times100=10\text{ \%}[/tex]Option D.
In the same way, there are 50 girls surveyed, and 12 play video games daily, so the percentage of girls that play video games daily is:
[tex]\frac{12}{50}\times100=0.24\times100=24\text{ \%}[/tex]Therefore, the only option that is not a reasonable interpretation of the data is C. because the percentage of boys that don't play video games daily is 10% instead of 5%
Which is the factored form of 3a2 - 24a + 48?а. (За — 8)(а — 6)b. 3a - 4)(a 4)c. (3a - 16)(a − 3)d. 3( -8)(a -8)
Ok, so:
We're going to factor this expression:
3a² - 24a + 48
First of all, we multiply and divide by 3 all the expression, like this:
3(3a² - 24a + 48) / 3
Now, we can rewrite this to a new form:
( (3a)² - 24(3a) + 144) / 3
Then, we have to find two numbers, whose sum is equal to -24 and its multiplication is 144.
And also we distribute:
((3a - 12 ) ( 3a - 12 )) / 3
Notice that the numbers we're going to find should be inside the brackets.
So, these numbers are -12 and -12.
Now, we factor the number 3 in the expression:
(3(a-4)*3(a-4))/3
And we can cancel one "3".
Therefore, the factored form will be: 3 (a - 4) (a - 4)
So, the answer is B.
I need help figuring out how to find sides a and b using the law of sine
Given the triangle ABC below.
a is the side facing b is the side facing
c is the side facing
We ara interested in calculating the value of side a and b.
To do this, we need to apply the "sine rule"
Sine rule state that
[tex]\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}[/tex]Where
a is the side facing b is the side facing
c is the side facing
To calculate b,
B = 95 , b = ?
C = 48, c=100
[tex]\begin{gathered} \frac{b}{\sin B}=\frac{c}{\sin C} \\ \frac{b}{\sin 95}=\frac{100}{\sin \text{ 48}} \\ \\ b\text{ x sin48=100 x sin95} \\ b=\frac{100\text{ x sin95}}{\sin 48} \\ b=134.05 \end{gathered}[/tex]b = 134 ( to nearest whole number)
To calculate a:
A = 37, a = ?
C = 48, c=100
[tex]\begin{gathered} \frac{a}{\sin A}=\frac{c}{\sin C} \\ \frac{a}{\sin37}=\frac{100}{\sin 48} \\ a\text{ x sin48 = 100 x sin37} \\ a=\frac{100\text{ x sin37}}{\sin 48} \\ a=80.98 \\ \end{gathered}[/tex]a = 81 ( to the nearest whole number)
why is height more specific measure than 'size'?
When you talk about height, it is a more precise value that require a vertical distance from a referenced point. While 'size' is not specific whatsoever and it's more like a form of range.
What is the first operation that should be performed to calculate (3 + 2) × 6÷5 - 4?
A) addition
B) division
C) subtraction
D) multiplication
Answer: A) addition
Step-by-step explanation:
because of BODMAS, you need to do the bracket first
help meeeeeeeeee pleaseee !!!!!
The value of the composite function is: (f o g)(2) = 33.
How to Find the Value of a Composite Function?To evaluate a composite function, take the following steps:
Step 1: Find the value of the inner function by substituting the value of x into the equation of the functionStep 2: Use the value of the output of the inner function as the input for the outer function and simplify to get the value of the composite function.Given the following:
f(x) = x² - 3x + 5
g(x) = -2x
Therefore,
(f o g)(2) = f(g(2))
Find the value of the inner function g(2):
g(2) = -2(2)
g(2) = -4
Find f(g(2)) by substituting x = -4 into the function f(x) = x² - 3x + 5:
(f o g)(2) = f(g(2)) = (-4)² - 3(-4) + 5
= 16 + 12 + 5
(f o g)(2) = 33
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given g(x)= -12f(x+1)+7 and f(-4)=2 fill in the blanks round answers to 2 decimal points as needed g( )=
We know the value of f(-4), which is 2
Let's think about a value of x in which we can calculate the value of f(x+1) using the given information (it means x+1 has to be equal to -4)
x+1=-4
x=-4-1
x=-5
Now use this value to calculate g(x)
[tex]\begin{gathered} g(-5)=-12\cdot f(-5+1)+7 \\ g(-5)=-12\cdot f(-4)+7 \end{gathered}[/tex]As we said, we already know the value of f(-4), use it to calculate g(-5)
[tex]\begin{gathered} g(-5)=-12\cdot2+7 \\ g(-5)=-24+7 \\ g(-5)=17 \end{gathered}[/tex](Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) And determine the quadrants of A+B and A-B.
Given that:
[tex]\cos A=\frac{5}{13}[/tex]Where:
[tex]0And:[tex]\cos B=\frac{3}{5}[/tex]Where:
[tex]0You need to remember that, by definition:[tex]\theta=\cos ^{-1}(\frac{adjacent}{hypotenuse})[/tex]Therefore, applying this formula, you can find the measure of angles A and B:
[tex]A=\cos ^{-1}(\frac{5}{13})\approx67.38\text{\degree}[/tex][tex]B=\cos ^{-1}(\frac{3}{5})\approx53.13\text{\degree}[/tex](a) By definition:
[tex]\sin \mleft(A+B\mright)=sinAcosB+cosAsinB[/tex]Knowing that:
[tex]\sin \theta=\frac{opposite}{hypotenuse}[/tex]You can substitute the known values into the equation in order to find the opposite side for angle A:
[tex]\begin{gathered} \sin (67.38\text{\degree)}=\frac{opposite}{13} \\ \\ 13\cdot\sin (67.38\text{\degree)}=opposite \\ \\ opposite\approx12 \end{gathered}[/tex]Now you know that:
[tex]\sin A=\frac{12}{13}[/tex]Using the same reasoning for angle B, you get:
[tex]\begin{gathered} \sin (53.13\text{\degree)}=\frac{opposite}{5} \\ \\ 5\cdot\sin (53.13\text{\degree)}=opposite \\ \\ opposite\approx4 \end{gathered}[/tex]Now you know that:
[tex]\sin B=\frac{4}{5}[/tex]Substitute values into the Trigonometric Identity:
[tex]\begin{gathered} \sin (A+B)=sinAcosB+cosAsinB \\ \\ \sin (A+B)=(\frac{12}{13})(\frac{3}{5})+(\frac{5}{13})(\frac{4}{5}) \end{gathered}[/tex]Simplifying, you get:
[tex]\begin{gathered} \sin (A+B)=\frac{36}{65}+\frac{20}{65} \\ \\ \sin (A+B)=\frac{36+20}{65} \end{gathered}[/tex][tex]\sin (A+B)=\frac{56}{65}[/tex](b) By definition:
[tex]\sin \mleft(A-B\mright)=sinAcosB-cosAsinB[/tex]Knowing all the values, you get:
[tex]\begin{gathered} \sin (A-B)=(\frac{12}{13})(\frac{3}{5})-(\frac{5}{13})(\frac{4}{5}) \\ \\ \sin (A-B)=\frac{36-20}{65} \\ \\ \sin (A-B)=\frac{16}{65} \end{gathered}[/tex](c) By definition:
[tex]\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A\cdot\tan B}[/tex]By definition:
[tex]\tan \theta=\frac{opposite}{adjacent}[/tex]Therefore, in this case:
- For angle A:
[tex]\tan A=\frac{12}{5}[/tex]- And for angle B:
[tex]\tan B=\frac{4}{3}[/tex]Therefore, you can substitute values into the formula and simplify:
[tex]\tan (A+B)=\frac{\frac{12}{5}+\frac{4}{3}}{1-(\frac{12}{5}\cdot\frac{4}{3})}[/tex][tex]\tan (A+B)=\frac{\frac{56}{15}}{1-\frac{48}{15}}[/tex][tex]\tan (A+B)=\frac{\frac{56}{15}}{-\frac{11}{5}}[/tex][tex]\tan (A+B)=-\frac{56}{33}[/tex](d) By definition:
[tex]\tan (A-B)=\frac{\tan A-\tan B}{1+\tan A\cdot\tan B}[/tex]Knowing all the values, you can substitute and simplify:
[tex]\tan (A-B)=\frac{\frac{12}{5}-\frac{4}{3}}{1+(\frac{12}{5}\cdot\frac{4}{3})}[/tex][tex]\tan (A-B)=\frac{\frac{16}{15}}{\frac{21}{5}}[/tex][tex]\tan (A-B)=\frac{16}{63}[/tex](e) Knowing that:
[tex]\sin (A+B)=\frac{56}{65}[/tex][tex]\tan (A+B)=-\frac{56}{33}[/tex]Remember the Quadrants:
By definition, in Quadrant II the Sine is positive and the Tangent is negative.
Since in this case, you found that the Sine is positive and the Tangent negative, you can determine that this angle is in the Quadrant II:
[tex]A+B[/tex]36. The widths of platinum samples manufactured at a factory are normally distributed, with a mean of 1.3 cm and a standard deviation of 0.3 cm. Find the z-scores that correspond to each of the following widths. Round your answers to the nearest hundredth, if necessary.(a) 1.7 cmz = (b) 0.9 cmz =
Part (a)
Using the formula for the z-scores and the information given, we have:
[tex]\begin{gathered} \text{ z-score=}\frac{\text{ data value }-\text{ mean}}{\text{ standard deviation }} \\ \text{ z-score=}\frac{1.7\text{ cm }-\text{ 1.3 cm}}{0.3\text{ cm}} \\ \text{ z-score=}\frac{0.4\text{ cm}}{0.3\text{ cm}}\text{ (Subtracting)} \\ \text{ z-score=1.33 (Dividing)} \\ \text{The z-score for 1.7 cm is 1.33 rounding to the nearest hundredth.} \end{gathered}[/tex]Part (b)
Using the formula for the z-scores and the information given, we have:
[tex]\begin{gathered} \text{ z-score=}\frac{\text{ data value }-\text{ mean}}{\text{ standard deviation }} \\ \text{z-score=}\frac{\text{ 0.9 cm }-1.3\text{ cm}}{\text{ 0.3 }}\text{ (Replacing the values)} \\ \text{z-score=}\frac{\text{ }-0.4}{\text{ 0.3 }}\text{ (Subtracting)} \\ \text{ z-score= }-1.33 \\ \text{The z-score for 0.9 cm is -1.33 rounding to the nearest hundredth.} \end{gathered}[/tex]The points (1,7) and (7,5) fall on a particular line. What is its equation in point-slope form?
Use one of the specified points in your equation. Write your answer using integers, proper fractions, and improper fractions. Simplify all fractions.
Answer:
[tex]y-7=-\dfrac{1}{3}(x-1)[/tex]
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{4.4cm}\underline{Slope Formula}\\\\Slope $(m)=\dfrac{y_2-y_1}{x_2-x_1}$\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ \\are two points on the line.\\\end{minipage}}[/tex]
To find the equation of a line that passes through two given points, first find its slope by substituting the given points into the slope formula.
Define the points:
(x₁, y₁) = (1, 7)(x₂, y₂) = (7, 5)Substitute the points into the slope formula:
[tex]\implies m=\dfrac{5-7}{7-1}=\dfrac{-2}{6}=-\dfrac{1}{3}[/tex]
Therefore, the slope of the line is -¹/₃.
[tex]\boxed{\begin{minipage}{5.8 cm}\underline{Point-slope form of a linear equation}\\\\$y-y_1=m(x-x_1)$\\\\where:\\ \phantom{ww}$\bullet$ $m$ is the slope. \\ \phantom{ww}$\bullet$ $(x_1,y_1)$ is a point on the line.\\\end{minipage}}[/tex]
To find the equation in point-slope form, simply substitute the found slope and one of the given points into the point-slope formula:
[tex]\implies y-7=-\dfrac{1}{3}(x-1)[/tex]
There were eight questions on Emily's math quiz, and she missed two questions.Which of the following diagrams represents the percentage of Emily's accuracy onthe quiz?A. 50%B. 75%C. 30%D. 10%
There are eight quartion in the quiz and two question missed. So Emily solved six question of the quiz.
Determine the accuracy of Emily.
[tex]\frac{6}{8}\times100=75[/tex]So Emily's accuracy is 75% and option B is correct.
Use the information given to find the equation of the line using the point-slope formula (y-y_1=m(x-x_1)). Then convert your answer to slope-intercept form (y=mx+b).(0,3) with a slope of 4The point slope form is (y-Answer)=Answer(x-Answer)Converting it to slope intercept form gives us y=Answerx+Answer
we have
m=4
point (0,3)
y-y1=m(x-x1)
substitute given values
y-3=4(x-0) ----> equation in point slope formConvert to slope-intercept form
y=mx+b
y-3=4x
adds 3 both sides
y=4x+3 ----> equation in slope-intercept formwhat is 0.024 ÷ 0.231
Answer:
0.10389610389
Step-by-step explanation:
Hi!
I plugged it into a calculator:
0.024 ÷ 0.231 = 0.10389610389
Have a great day! :)
I inserted a picture of the question, please give a VERY SHORT explanation
To multiply two polynomials we have to multiply each term in one polynomial by each term in the other polynomial, and then add the similar terms, as follows:
Write the equation of the circle centered at (−4,−2) that passes through (−15,19)
In this problem, we are going to find the formula for a circle from the center and a point on the circle. Let's begin by reviewing the standard form of a circle:
[tex](x-h)^2+(y-k)^2=r^2[/tex]The values of h and k give us the center of the circle, (h,k). The value r is the radius. We can begin by substituting the values of h and k into our formula.
Since the center is at (-4, -2), we have:
[tex]\begin{gathered} (x-(-4))^2+(y-(-2))^2=r^2 \\ (x+4)^2+(y+2)^2=r^2 \end{gathered}[/tex]Next, we can use the center and the given point on the circle to find the radius.
Recall that the radius is the distance from the center of a circle to a point on that circle. So, we can use the distance formula:
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]Let
[tex](x_1,y_1)=(-4,-2)[/tex]and let
[tex](x_2,y_2)=(-15,19)[/tex]Now we can substitute those values into the distance formula and simplify.
[tex]\begin{gathered} r=\sqrt{(-15-(-4))^2+(19-(-2))^2} \\ r=\sqrt{(-11)^2+(21)^2} \\ r=\sqrt{562} \end{gathered}[/tex]Adding that to our equation, we have:
[tex]\begin{gathered} (x+4)^2+(y+2)^2=(\sqrt{562})^2 \\ (x+4)^2+(y+2)^2=562 \end{gathered}[/tex]Can anyone please help me with this fast? Thank you!
Answer:
Step-by-step explanation:
16. 4/16 1/16 1/16 or 6/16
17. 1/16 1/16 or 2/16
18. 7/16 1/16 2/16 or 10/16
19. 2/16
20 4/16 1/16 1/16 7/16 1/16 2/16 or 16/16=1
Draw the following vectors using the scale 1 cm = 50 km/h. Plant the tail at the origin. A. 200 km/h on a bearing of 020° B. 75 km/h S 10° W C. 350 km/h NE
Solution
a)
200 km/h on a bearing of 020°
Scale 1 cm = 50 km/h.
[tex]Length\text{ = }\frac{200}{50}\text{ = 4cm}[/tex]b)
B. 75 km/h S 10° W
[tex]Lenght\text{ = }\frac{75}{50}\text{ = 1.5cm}[/tex]C. 350 km/h NE
[tex]Length\text{ = }\frac{350}{50}\text{ = 7cm}[/tex]у A 5 8 106 С C m2l= m22= m23= mZ4= m25= needing quadrilaterals area
Angles in a quadrilaterals
The sum of all interior angles in a quadrilateral is 360°
Angle 5 is congruent with angle of 106°
Thus measure of 5 = 106°
These two angles add up to 212°. The remaining to reach 360° is:
360° - 212° = 148°
Angles 1, 2, 3, and 4 are congruent, thus the measure of each one of them is 148/4=37°. Thus
measure of 1 = measure of 2 = measure of 3 = measure of 4 = 37°
Lee Ann is planning a bridal shower for her best friend. At the party, she wants to serve 3 beverages, five appetizers, and three desserts, but she doesn't not have time to cook. She can choose from 9 bottle drinks, 9 Frozen appetizers, and 12 prepared desserts at the supermarket. How many different ways can lie and pick up the food and drinks to serve at the bridal shower?
2328480 ways
ExplanationWe want to find out how many ways there are to select an item. The combination formula is a formula to find the number of ways of picking "r" items from a total of "n" items.
This number is given by:
[tex]undefined[/tex]write 2500g in appropriate prefix pls.
Answer: 2.5kg
Step-by-step explanation:
I am assuming you mean to simplify it. So 2.5kg
1g=1000kg
2500/1000=2.5
in the diagram segment AD and AB are tangent to circle C solve for x
A property ostates that if two lines that are tangent to the circle intersect in an external point, they are congruent, i.e. they have the same length.
[tex]\begin{gathered} AD=AB \\ x^2+2=11 \end{gathered}[/tex]From this expression we can determine the possible values of x. The first step is to equal the expression to zero
[tex]\begin{gathered} x^2+2-11=11-11 \\ x^2+2-11=0 \\ x^2-9 \end{gathered}[/tex]The expression obtained is a quadratic equation, using the queadratic formula we can determine the possible values of x:
[tex]\begin{gathered} f(x)=ax^2+bx+c \\ x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \end{gathered}[/tex]For our expression
[tex]x^2+0x+-9[/tex]The coefficients are
a=1
b=0
c=-9
Replace them in the formula
[tex]\begin{gathered} x=\frac{-0\pm\sqrt[]{0^2-4\cdot1\cdot(-9)}}{2\cdot1} \\ x=\frac{0\pm\sqrt[]{36}}{2} \\ x=\frac{0\pm6}{2} \end{gathered}[/tex]Now calculate both possible values:
Positive:
[tex]\begin{gathered} x=\frac{+6}{2} \\ x=3 \end{gathered}[/tex]Negative:
[tex]\begin{gathered} x=\frac{-6}{2} \\ x=-3 \end{gathered}[/tex]The possible values of x are 3 and -3
Enter a rule for each function f and g, and then compare their domains, ranges, slopes, and y-intercepts.The function f(x) has a slope of -2 and has a y-intercept of 3. The graph shows the function g(x).
The rule of the function f(x) is : -2x + 3
To find the rule of the function g(x) let's calculate the slope of the line
[tex]m=\frac{y2-y1}{x2-x1}=\frac{-11-5}{4-0}=\frac{-16}{4}=-4[/tex]The slope of the line is -4 and the intercept is 5 ( from the graph).
The rule of the function g(x) is : -4x + 5
The domains of f(x) and g(x) is All real numbers, because there is not any number of x which doesn't have a corresponding y-coordinate.
The ranges of f(x) and g(x) is All real numbers, because there is not any number of y which doesn't have a corresponding x-coordinate.
The slope of f(x) is greater than g(x) (-2 is greater than -4)
The y-intercept of f(x) is less than the y-intercept of g(x).(3 is less than 5)
Find y if the line through (1, y) and (8, 2) has a slope of 3.
Answer: -19
Step-by-step explanation:
I think I am correct I am sorry if not.
Here is how I got it-
m = 21 / 7 = 3 / 1 = 3
Equation: y = 3x - 22
Answer:
y = -19
Step-by-step explanation:
Pre-SolvingWe are given two points: (1, y) and (8,2).
We want to find the value of y if the slope of the line is 3.
SolvingThe slope (m) can be calculated from two points using the formula [tex]m = \frac{y_2-y_1}{x_2-x_1}[/tex] where [tex](x_1, y_1)[/tex] and [tex](x_2, y_2)[/tex] are points
We can label the values of the points to avoid any confusion and mistakes.
[tex]x_1 = 1\\y_1=y \\x_2=8\\y_2=2[/tex]
Substitute these values into the formula.
[tex]m = \frac{y_2-y_1}{x_2-x_1}[/tex]
[tex]m = \frac{2-y}{8-1}[/tex]
Remember that the slope of the line is 3, so we can substitute m as 3.
Replace m as 3.
[tex]3 = \frac{2-y}{8-1}[/tex]
Subtract.
[tex]3 = \frac{2-y}{7}[/tex]
Multiply both sides by 7.
[tex]3 * 7 = 7(\frac{2-y}{7})[/tex]
21 = 2-y
Subtract 2 from both sides.
19 = -y
Divide both sides by -1.
-19 = y
y = - 19.
Sara spent 35 minutes on math homework and 20 minutes on reading homework. Mia spent a total of 40
minutes on reading and math homework. How much longer did Sara spend on her homework than Mia?
Sara spent 15 minutes longer than (the difference is 15 min) Mia in her homework.
According to the question,
We have the following information:
Sara spent 35 minutes on math homework and 20 minutes on reading homework. Mia spent a total of 40 minutes on reading and math homework.
So, it means that the total time spent by Sara in her homework is:
35+20 minutes
55 minutes
So, the differences between their time spent in her homework (will give us the more time taken by Sara) is:
Time spent by Sara in her homework-time spent by Mia in her homework
(55-40) minutes
15 minutes
Hence, Sara spent 15 more minutes than Mia.
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Fill in the table using this function rule.
Answer:
-8, 2, 12, 22
Step-by-step explanation:
[tex]y = 5x+2\\y = 5(-2)+2\\y=-10+2\\y=-8[/tex]
[tex]y = 5x+2\\y = 5(0)+2\\y=0+2\\y=2[/tex]
[tex]y = 5x+2\\y = 5(2)+2\\y=10+2\\y=12[/tex]
[tex]y = 5x+2\\y = 5(4)+2\\y=20+2\\y=22[/tex]
How to find the (r) or difference in this scenario:
Aliens Away is a new video game where a player must eliminate a certain number of aliens on the screen by scaring them with an adorable house cat. When James plays the game, he eliminates 64 aliens in the first level and 216 aliens in the fourth level. If the number of aliens are destroyed in a geometric sequence from one level to the next, how many total aliens will James have wiped out by the end of the sixth level?
It is given that it is a geometric sequence, if I am not mistaken it is the explicit formula.
IF YOU COULD PLASE EXPLAIN:)
Total number of aliens James have wiped out by the end of the sixth level is 1330
The number of aliens eliminated in first level a = 64
The number of aliens eliminated in the fourth level = 216
The sequence is in geometric sequence
The nth term of the geometric sequence is
[tex]a_n=ar^{n-1}[/tex]
The fourth term is 216
216 = [tex]64r^3[/tex]
[tex]r^{3}[/tex] = 216/64
r = 3/2
Then we have to find the total alien James killed by the end of sixth level
Sum of n terms = [tex]\frac{a(r^n-1)}{r-1}[/tex]
Substitute the values in the equation
= [tex]\frac{64(1.5^6-1)}{1.5-1}[/tex]
= 665/0.5
= 1330
Hence, total number of aliens James have wiped out by the end of the sixth level is 1330
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200 lottery tickets are sold for $6 each. The person with the single winning ticket will get $71. What is the expected value for a ticket in this lottery?
Given:
200 lottery tickets are sold for $6 each.
The person with the single winning ticket will get $71.
So, The probability of winning = 1/200
The probability of losing =
[tex]undefined[/tex]
Answer: the expected value is. aroud 1-2
Step-by-step explanation:
Find the measure of all the angles if m<2 = 76°
By opposite angles we know that:
[tex]\begin{gathered} m1\measuredangle=m3\measuredangle \\ m2\measuredangle=m4\measuredangle \\ m5\measuredangle=m7\measuredangle \\ m8\measuredangle=m6\measuredangle \end{gathered}[/tex]By corresponding angles we know that
[tex]\begin{gathered} m5\measuredangle=m1\measuredangle \\ m2\measuredangle=m6\measuredangle \\ m4\measuredangle=m8\measuredangle \\ m7\measuredangle=m3\measuredangle \end{gathered}[/tex]by complementary angles we know that
[tex]\begin{gathered} m1\measuredangle+m2\measuredangle=180º \\ m1\measuredangle+76º=180º \\ m1\measuredangle=104º \end{gathered}[/tex]Using the correspondence and opposite angles:
[tex]\begin{gathered} m1\measuredangle=m3\measuredangle=m5\measuredangle=m7\measuredangle=104º \\ m2\measuredangle=m4\measuredangle=m6\measuredangle=m8\measuredangle=76º \end{gathered}[/tex]Figure R = figure R". Describe a sequence of three transformations you canperform on figure R to show this. Show your work.
1) Rotate the figure 90º clockwise.
To rotate a figure 90º clockwise you have to perform the following transformation:
(x,y)→(y,-x)
So for each point of the figure you have to swich places between x and y.
And you have to multiply the x coordinate by -1.
R has 6 points:
(-4,-5)
(-7,-5)
(-7,-4)
(-5,-4)
(-6,-3)
(-2,-2)
First step: swich places between the coordinates of each point:
(x,y) → (y,x)
(-4,-5)→(-5,-4)
(-7,-5)→(-5,-7)
(-7,-4)→(-4,-7)
(-5,-4)→(-4,-5)
(-6,-3)→(-3,-6)
(-2,-2)→(-2,-2)
Second step, multiply the y-coordinate of the new set by -1
(y,x)→ (y,-x)
(-5,-4)→ (-5,4)
(-5,-7)→ (-5,7)
(-4,-7)→ (-4,7)
(-4,-5)→ (-4,5)
(-3,-6)→ (-3,6)
(-2,-2)→ (-2,2)
After the rotation, the figure moved from the 3rth quadrant to the 2nd quadrant.
Solve the equation on the interval [0, 2\small \pi). Show all work. Do not use a calculator - use your unit circle!
SOLUTION
Write out the equation given
[tex]\cos ^2x+2\cos x-3=0[/tex]Let
[tex]\text{Cosx}=P[/tex]Then by substitution, we obtain the equation
[tex]p^2+2p-3=0[/tex]Solve the quadractic equation using factor method
[tex]\begin{gathered} p^2+3p-p-3=0 \\ p(p+3)-1(p+3)=0 \\ (p-1)(p+3)=0 \end{gathered}[/tex]Then we have
[tex]\begin{gathered} p-1=0,p+3=0 \\ \text{Then} \\ p=1,p=-3 \end{gathered}[/tex]Recall that
[tex]\cos x=p[/tex]Hence
[tex]\begin{gathered} \text{when p=1} \\ \cos x=1 \\ \text{Then } \\ x=\cos ^{-1}(1)=0 \\ \text{hence } \\ x=0 \end{gathered}[/tex]Similarly,
[tex]\begin{gathered} \text{When p=-3} \\ \cos x=-3 \\ x=\cos ^{-1}(-3) \\ x=no\text{ solution} \end{gathered}[/tex]Therefore x=0 is the only valid solution on the given interval [0,2π).
Answer; x=0
(2i) - (11+2i) complex numbers