Tan(theta)= 12/5, sin(theta) > 0Trig: find the exact values of the remaining trigonometric functions of (theta) satisfying the given conditions. If it it undefined type UNDEFINEDSIN(theta)=Cos(theta)=Csc(theta)=Sec(theta)=Cot(theta)=
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Answer:
The exact values of the remaining trigonometric functions of (theta) satisfying the given conditions are;
[tex]\begin{gathered} \sin \theta=\frac{12}{13} \\ \cos \theta=\frac{5}{13} \\ \text{csc}\theta=\frac{13}{12} \\ \text{sec}\theta=\frac{13}{5} \\ \text{cot}\theta=\frac{5}{12} \end{gathered}[/tex]Explanation:
Given the above conditions;
[tex]\begin{gathered} \tan \theta=\frac{12}{5} \\ \sin \theta>0 \end{gathered}[/tex]The only quadrant where the value of tangent and sine is positive is the first quadrant.
So, the value of cosine will also be positive since in the first quadrant sine, cosine and tangent are positive (greater than zero).
Given;
[tex]\begin{gathered} \tan \theta=\frac{12}{5}=\frac{opposite}{\text{adjacent}} \\ So;\text{ from trigonometry} \\ \text{opposite a = 12} \\ \text{adjacent b = 5} \end{gathered}[/tex]We need to calculate the hypotenus using pythagoras theorem;
[tex]\begin{gathered} c^2=a^2+b^2 \\ c^2=12^2+5^2 \\ c^2=144+25=169 \\ c=\sqrt{169} \\ c=13 \\ \text{Hypotenuse c = 13} \end{gathered}[/tex]we can use this values to find the exact values of the following;
[tex]\begin{gathered} \sin \theta=\frac{opposite}{hypotenuse}=\frac{12}{13} \\ \cos \theta=\frac{adjacent}{hypotenuse}=\frac{5}{13} \\ \text{csc}\theta=\frac{1}{\sin \theta}=\frac{13}{12} \\ \text{sec}\theta=\frac{1}{\cos \theta}=\frac{13}{5} \\ \text{cot}\theta=\frac{1}{\tan \theta}=\frac{5}{12} \end{gathered}[/tex]Therefore, the exact values of the remaining trigonometric functions of (theta) satisfying the given conditions are;
[tex]\begin{gathered} \sin \theta=\frac{12}{13} \\ \cos \theta=\frac{5}{13} \\ \text{csc}\theta=\frac{13}{12} \\ \text{sec}\theta=\frac{13}{5} \\ \text{cot}\theta=\frac{5}{12} \end{gathered}[/tex]
Related Questions
I just need help simplifying it into a single exponential expression. Thank you
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Answer
x = -½
Explanation
To answer this, we will use the laws of indices to solve the left hand side
When the same variable, carrying different powers are multiplied with each other, then, the result is that same variable raised to the power of the sum of the two powers from before multiplication.
[tex]\begin{gathered} e^xe^{x+1}=1 \\ e^{x+x+1}=1 \\ e^{2x+1}=1 \end{gathered}[/tex]Take the natural logarithms of both sides
[tex]\begin{gathered} e^{2x+1}=1 \\ In(e^{2x+1})=In(1) \\ (2x+1)(\text{In e) = In 1} \\ In\text{ e = 1, In 1 = 0} \\ (2x+1)(1)=0 \\ 2x+1=0 \end{gathered}[/tex]2x + 1 = 0
2x = -1
Divide both sides by 2
(2x/2) = (-1/2)
x = (-1/2)
x = -½
Hope this Helps!!!
Combine like terms. 8b + 6b − 16 = ?
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EXPLANATION
Considering the equation:
8b + 6b - 16
Combining like terms:
If a circle has an area of 31,400 inches what is its circumference
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Given:
If a circle has an area of 31,400 inches.
Required:
what is its circumference
Explanation:
We know the area of circle formula
[tex]\begin{gathered} \pi r^2=31400 \\ r=99.97 \end{gathered}[/tex]Also, circumference of circle
[tex]\begin{gathered} =2\pi r \\ =2\times\pi\times99.97 \\ =628.13\text{ in} \\ =628\text{ in \lparen approx.\rparen} \end{gathered}[/tex]Answer:
Option A is correct.
Find the annual percentage yield for an investment at the following rates. (Round your answers to two decimal places.)
(a) 7.8% compounded monthly
(b) 8% compounded continuously
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The annual percentage yield for an investment at 7.8% compounded monthly rate is 8.08% and the annual percentage yield for an investment a 8% compounded continuously rate is 8.33%.
a.
we know that,
Formula for finding annual percentage yield based on compounded monthly is:
Annual percentage yield (APY) =[tex](1 + r/n)^{n}[/tex] - 1
Here
r = interest rate = 7.8%
n = number of times interest is compounded in year = 12
substitute r and n and we get,
APY = (1 + 7.8%/12)^12 - 1
APY = 8.08 %.
b.
when interest is compounded continuously , finding the APY using the below formula:
APY = [tex]e^{rt}[/tex] - 1
e = constant = 2.7182818
r = interest rate = 8%
t = 1(APY is measuring continuously)
APY = 2.7182818^(8%*1)
APY = 8.33%.
Therefore the annual percentage yield for an investment at 7.8% compounded monthly rate is 8.08% and the annual percentage yield for an investment a 8% compounded continuously rate is 8.33%.
Learn more about the annual percentage yield here:
https://brainly.com/question/27997520
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The vertices of AJKL are J(1.3), K(4,4), and L(3,1). Graph AJKL and its image after a reflection in the line y = x.
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Answers:
J ' (3, 1)K ' (4, 4)L ' (1, 3)The graph is shown below=======================================================
Explanation:
The rule to reflect over the line y = x is this
[tex](\text{x},\text{y})\to(\text{y},\text{x})[/tex]
The x and y coordinates swap places. A point like (5,7) moves to (7,5) as an example.
When we apply that rule to the points J, K, L, we get the following
J(1,3) moves to J ' (3,1)K(4,4) stays at (4,4) so K ' (4, 4)L(3,1) moves to L ' (1,3)Then take note of these two facts:
Point K and point K' are at the same location. In other words, point K doesn't move. Points on the mirror line will not move when applying reflections.Points J and L swap places since both involve coordinates 1 and 3 in either order.Because of these two facts, triangle JKL and triangle J'K'L' occupy the same exact space and area. All that's happened really is almost all of the vertices have been relabeled to something else.
See below for the graph.
valent systems discussionNGSubre3x - 4y - 10Add a Comment:6x + y + 38and-9x + 12y = -30Media Comment9x - 3y = 48SaveThese systems are said to be equivalent Both of the equations in the secondsystem came from the first system somehow.Two questions: How was the first equation in the second system formed fromthe first system? And how was the second equation in the second systemformed from the first system?
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So, we have the two following equation systems:
I. a.) 3x - 4y = 10
I. b.) 6x + y = 38
and
II. a.) -9x + 12y = -30
II. b.) 9x - 3y = 48
We know that these two systems are equivalent, which means that each equation from the second system can be formed as a linear combination of the other two equations from the first system.
From the equation II. a. we have that (with z and w being integer numbers)
z*3x + w*6x = -9x or 3z + 6w = -9
z*(-4y) + w*y = 12y or -4z + w = 12
z*10 + w* 38 = -30 or 10z + 38w = -30
Solving that system for z and w, we discover that z = -3 and w = 0, wich means that equation II. a. is formed by all terms from equation I. a. multiplied by -3.
Doing the same for equation II. b., we have:
z*3x + w*6x = 9x or 3z + 6w = 9
z*(-4y) + w*y = -3y or -4z + w = -3
z*10 + w* 38 = 48 or 10z + 38w = 48
Solving that system for z and w, we discover that z = 1 and w = 1, wich means that equation II. b. is formed by the sum of all terms from the equations I. a. and I. b.
What is the probability of a hiker seeing a finch???
Answers
Answer
B. 1/3
Explanation
From the value table; it was given that:
The number of finch = 20
The total number birds observed = 60
Therefore, the probability of a hiker seeing a finch will be:
[tex]P(hiker\text{ }seeing\text{ }a\text{ }finch)=\frac{number\text{ }of\text{ }finch}{Total\text{ }number}=\frac{20}{60}=\frac{1}{3}[/tex]Hence, the correct answer is option B. 1/3
if one edge of a cube is 12 cm what is its volume
Answers
Answer:
1728 cm³
Explanation:
The volume of a cube can be calculated as:
[tex]V=a^3[/tex]Where a is the length of the edge. So, if we replace a by 12 cm, we get that the volume of the cube is:
[tex]\begin{gathered} V=(12cm)^3 \\ V=12\operatorname{cm}\times12\text{ cm }\times12\text{ cm} \\ V=1728cm^3 \end{gathered}[/tex]Therefore, the volume of the cube is 1728 cm³
A system of equations is show. Y=-1/2xX^2+y^2=20Which of the following ordered pairs are solutions of the system of equations?Select two correct ordered pairs. A. (8,-4)B. (-4,2)C. (0,0)D. (4,-2)E. (8,-4)
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Explanation:
For the system of equations given below
[tex]\begin{gathered} y=-\frac{1}{2}x \\ x^2+y^2=20 \end{gathered}[/tex]If we are to solve, we will simply substitute y=-1/2x into the second equation, so that
[tex]\begin{gathered} x^2+(-\frac{1}{2}x)^2=20 \\ x^2+\frac{1}{4}x^2=20 \\ \frac{4x^2+1x^2}{4}=\frac{5}{4}x^2=20 \\ \text{cross multiply} \end{gathered}[/tex][tex]\begin{gathered} x^2=\frac{20\times4}{5} \\ x^2=16 \\ x=-4 \\ \text{and} \\ x=4 \end{gathered}[/tex]The next step will be to substitute x=4 and x=-4 into the equation to get the required values of y
When x=4
[tex]\begin{gathered} y=-\frac{1}{2}x \\ y=-\frac{1}{2}(4)=-2 \\ \text{Thus one of the pair is} \\ (4,-2) \end{gathered}[/tex]Also,
When x=-4
[tex]\begin{gathered} y=-\frac{1}{2}x \\ y=-\frac{1}{2}(-4) \\ y=2 \\ \text{Thus, the second pair is} \\ (-4,2) \end{gathered}[/tex]Hence, the answers are: Options B and D
What is 2*89+56(7659)
Answers
Answer:
[tex]2\times89+56(7659)=429,082[/tex]Explanation:
Given the expression:
[tex]2\times89+56(7659)[/tex]First of all, remove the parentheses
[tex]2\times89+428904[/tex]Perform the multiplication
[tex]178+428904[/tex]Add
[tex]429,082[/tex]If 3x + 7 = 2x + 10, what is 4x?
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First let's find the value of x by solving this equation:
[tex]\begin{gathered} 3x+7=2x+10 \\ 3x-2x=10-7 \\ x=3 \end{gathered}[/tex]Now that we have the value of x, we can calculate the value of 4x:
[tex]4x=4\cdot3=12[/tex]So the value of 4x is 12.
Translate the following phrase into a math expression or equation whichever is appropriate use x to represent the unknown number A number increased by three nineteenths of itself
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Answer:
[tex]x+\frac{3}{19}x[/tex]We have an unknown number x. We want to increase that number by 3/19 of itself. This means, that we must add 3/19 of x to x. This can we writted as:
[tex]x+\frac{3}{19}x[/tex]determine the number of cubic centimeters of ice cream that you have in the cone and the scoop on top
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Given a cone and a scoop of ice cream.
To determine: The cubic centimeters of ice cream
Solution:
The ice is a combination of the shape of a cone and a hemisphere. To find the cubic centimeters of the ice cream, we would find the volume of the cone and the hemisphere.
[tex]V_{\text{ice cream}}=V_{cone}+V_{hemisphere}[/tex][tex]\begin{gathered} V_{\text{cone}}=\frac{1}{3}\pi r^2h \\ d=6\operatorname{cm}(given) \\ r=\frac{d}{2}=\frac{6\operatorname{cm}}{2}=3\operatorname{cm} \\ h=13\operatorname{cm}(\text{given)} \end{gathered}[/tex][tex]\begin{gathered} V_{\text{cone}}=\frac{1}{3}\pi r^2h \\ V_{\text{cone}}=\frac{1}{3}\pi\times3^2\times13 \\ V_{\text{cone}}=39\pi\operatorname{cm}^3 \end{gathered}[/tex][tex]\begin{gathered} V_{\text{hemisphere}}=\frac{2}{3}\pi r^3 \\ r=3\operatorname{cm} \\ V_{\text{hemisphere}}=\frac{2}{3}\pi\times3^3 \\ V_{\text{hemisphere}}=18\pi cm^3 \end{gathered}[/tex][tex]\begin{gathered} V_{\text{ice cream}}=V_{cone}+V_{hemisphere} \\ V_{\text{ice cream}}=39\pi cm^3+18\pi cm^3 \\ V_{\text{ice cream}}=57\pi cm^3 \\ V_{\text{ice cream}}=179.07\operatorname{cm}^3 \end{gathered}[/tex]Hence, the number of cubic centimeters of ice cream is 179.07cm³
Tessa works 3 days a week. She works 4.5 hours each day.Tessa wants to know how many hours she works each week.Tessa draws the area model below to find the productof 3 and 4.5.40.533 X 4 = ?3 X 0.5 = ?How many hours does Tessa work each week? Write your answer in the blank.1.8 hoursAGс
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Solution
[tex]\begin{gathered} 3\times4=12 \\ \\ 3\times0.5=1.5=\frac{3}{2} \end{gathered}[/tex]- Thus, Tessa worked for
[tex](12+1.5)hours=13.5hours[/tex]Final Answer
Tessa worked for 13.5 hours
Please help me solve this problem : b and c
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Solution:
(b)
The amount of mineral fuels, including oil and vehicle is;
[tex]99.3+60.5=159.8[/tex]The total amount of exports is;
[tex]450.7[/tex]Thus, the percentage relative of mineral fuels, including oil and vehicle to total exports is;
[tex]\frac{159.8}{450.7}\times100=35.5[/tex](c) The total amount of top five Canadian exports is;
[tex]99.3+60.5+34.5+18.3+14.3=226.9[/tex]Thus, the percentage relative of mineral fuels, including oil and vehicle to top five exports is;
[tex]\frac{159.8}{226.9}\times100=70.4[/tex]Find the measure of the supplement for the angle 78
Answers
Two angles are supplementary if their sum is equal to 180°; therefore, in our case, if [tex]\begin{gathered} \angle x+78\degree=180\degree \\ \Rightarrow\angle x=180\degree-78\degree=102\degree \end{gathered}[/tex]Therefore, the answer is 102°
Diameters of a country's currency coins are normally distributed with a mean of 2 cm and standard deviation of 0.5 cm.If Aiden has a coin that is at the 40th percentile, what diameter does Aiden's coin measure to? Round your answer to onedecimal place. Do not include units in your answer.
Answers
The 40 percentile means that we are looking for the diameter of the coin (X) that separates the bottom 40% of the distribution from the top 60%.
Since we have to work with the standard normal distribution we have:
[tex]P(Z\le z)=0.4[/tex]Now we use the given table to determine the value of "z" that is closest to the probability of 0.4.
The way to read the table is the following. The first column tells you the first decimal digit of the z-value and the first row tells the second digit.
From the table we see that the closest number to the probability of 40% or 0.4 is 0.4013, this means that the z-value is z = -0.25. Now using the z-value formula:
[tex]z=\frac{X-\mu}{\sigma}[/tex]Since X = d, we get:
[tex]z=\frac{d-\mu}{\sigma}[/tex]Solving for "d" first by multiplying both sides by sigma:
[tex]z\sigma=d-\mu[/tex]Adding the mean to both sides:
[tex]z\sigma+\mu=d[/tex]Replacing the given values:
[tex](-0.25)(0.5)+2=d[/tex]Solving the operations:
[tex]1.875=d[/tex]Therefore, the value of "d" is 1.875 cm. Rounded would be 1.9 cm.
13. What is the slope of the following points?(6, 7), (-2, 7)
Answers
To find the slope of the points (6, 7), (-2, 7)
we will use the formula;
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]From the points given;
x₁=6 y₁=7 x₂=-2 y₂=7
substituting into the formula:
[tex]m=\frac{7-7}{-2-6}[/tex][tex]m=\frac{0}{-8}=0[/tex]The slope is 0
Tomorrow, there is a 40% chance of rain. Also, tomorrow, there is a 40% chance ofhaving a test you forgot about. What are the odds of tomorrow being a bad day (rainand forgotten test)?
Answers
We are dealing with two independent events. This is so because the chance or probability that it it will rain is independent of the chance or probability that there would be a test that you forgot about. If P(A) represents the event that it rains and P(B) represents the event that there is a test that you forgot about, then from the information given,
P(A) = 40/100 = 0.4
P(B) = 40/100 = 0.4
Thus, the odds or probability of tomorrow being a bad day(rain and forgotten test) is P(A) * P(B). Thus,
P(A) * P(B) = 0.4 * 0.4 = 0.16
The answer is 0.16
Translate the sentence into an equation. Eight times the sum of a number and 3 is 6.Use the variable y for the unknown number.
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Let the number be y
8(y + 3) = 6
Can someone please help me figure out this math problem? I don't understand how they're wanting it to be worked out.
Answers
We will have the following:
R 60 Cups
B 20 Cups
Y 30 Cups
O = 5Y:3R
P = 7R:2B
Here R, B, Y, O & P are red, blue, yellow, orange, and purple respectively.
Now, we proceed as follows:
[tex]5Y+3R=O[/tex]and:
[tex]7R+2B=P[/tex][tex]R+B+Y\le110[/tex]We will also have:
35/3Y:7R:2B [This is the proportional relationship of the base colors at any given time under the constraints of the mixing of colors].
35/3Y + 7R + 2B
This means that for each cup of yellow paint we will be able to use 1/2 cup to mix.
Now, we solve the system:
R B Y
7 : 2 : 35/3 = 62/3
*v1 = (110)/(62/3) = 165/31
**v2 =
On two investments totaling $9,000, Lydia lost 4% on one and earned 6 % on the other. If her net annual receipts were $170, how muchwas each investment?
Answers
Answer
The investment at -4% is $3700
The investest made at +6% is $5300
Step-by-step explanation:
Given
The total amount of the two investments = $9000
She lost 4% on one and gained 6% on the other
Let x represents the first investment
Let y represents the second investment
Since the two investments make a total of $9000
This can be expressed mathematically as
x + y = 9000 ---------------- equation 1
If she lost 4% on one of the investments and earned 6% in the other investment
Then this can be expressed as
-0.04x + 0.06y = 170
-0.04x + 0.06y = 170 ---------- equation 2
These two systems of the linear equation can be solved simultaneously
x + y = 9000 ------------- equation 1
-0.04xx + 0.06y = 170 ---- equation 2
We will be using the substitution method to solve the above equations
Make the x the subject of the formula in equation 1
x = 9000 - y
Substitute the value of x into equation 2
-0.04(9000 - y) + 0.06y = 170
Open the parentheses
-360 + 0.04y + 0.06y = 170
Collect the like terms
0.04y + 0.06y = 170 + 360
0.1 y = 530
Divide both sides by 0.1
y = 530 / 0.1
y = $5300
From equation 1
x + y = 9000
y = $5300
x = 9000 - y
x = 9000 - 5300
x = $3700
find the general equation of the ellipse passing though points A and B, center at C.
Answers
The general equation of an ellipse is
[tex]\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1[/tex]Where (h, k) are the coordinates of the center.
For our ellipse, we already have the center (2, 1), then, our equation will be
[tex]\frac{(x-2)^2}{a^2}+\frac{(y-1)^2}{b^2}=1[/tex]Using the points that belong to this ellipse, we can substitute them on this equation to find the coefficients a and b.
First, let's use the point (4, 1). Doing the substitution, we have
[tex]\begin{gathered} \frac{(4-2)^2}{a^2}+\frac{(1-1)^2}{b^2}=1 \\ \frac{(2)^2}{a^2}=1 \\ a=2 \end{gathered}[/tex]Using the other point, we can calculate the other coefficient.
[tex]\begin{gathered} \frac{(1-2)^2}{4}+\frac{(1+2\sqrt[]{3}-1)^2}{b^2}=1 \\ \frac{(-1)^2}{4}+\frac{(2\sqrt[]{3})^2}{b^2}=1 \\ \frac{1}{4}+\frac{12^{}}{b^2}=1 \\ \frac{12^{}}{b^2}=\frac{3}{4} \\ 4\cdot12=3\cdot b^2 \\ b^2=16 \\ b=4 \end{gathered}[/tex]And then, we have our ellipse equation.
[tex]\frac{(x-2)^2}{4}+\frac{(y-1)^2}{16}=1[/tex]Find the value of x, given the image below.Select one:a.Cannot be determinedb.28c.16d.6.9
Answers
x = 6.9 (option D)
Explanation:Given:
DC = 4, CA = 7
CE = x, BA = x + 12
To find:
the value of x
To determine x, we will apply the similarity theorem for triangles:
For two triangles to be similar, the ratio of corresponding sides will be equal
Triangle EDC is similar to triangle BDA
[tex]\begin{gathered} DC\text{ corresponds to DA} \\ CE\text{ corresponds to AB} \\ \\ The\text{ ratio:} \\ \frac{DC}{DA}\text{ = }\frac{CE}{AB} \end{gathered}[/tex][tex]\begin{gathered} DA\text{ = DC +}CA \\ DA\text{ = 4 + 7 = 11} \\ \\ substitute\text{ the values:} \\ \frac{4}{11}=\frac{x}{x\text{ + 12}} \\ cross\text{ multiply:} \\ 4(x\text{ + 12\rparen = 11\lparen x\rparen} \\ 4x\text{ + 48 = 11x} \end{gathered}[/tex][tex]\begin{gathered} collect\text{ like terms:} \\ 48\text{ = 11x - 4x} \\ 48\text{ = 7x} \\ \\ divide\text{ both sides by 7:} \\ \frac{48}{7}\text{ = }\frac{7x}{7} \\ x\text{ = 6.857} \\ \\ x\text{ = 6.9 \lparen1 decimal place\rparen \lparen option D\rparen} \end{gathered}[/tex]A sixth-grade science club needs $180 to pay for the tickets to ascience museum. All tickets cost the same amount.What could 180 ÷ 15 mean in this situation? Describe two differentpossible meanings of this expression. then, find the quotient and explain what it means in each case
Answers
since the total cost is given $180, then, this cost can be found through the product between the number of students in sixth grade and the cost per ticket at the science museum.
then, if 15 were the number of students in sixth grade you could find the cost per ticket at the museum through the quotient
[tex]\begin{gathered} \frac{180}{15}=\text{price per ticket} \\ 12=\text{price per ticket } \end{gathered}[/tex]you if 15 was the cost per ticket at the museum youcould find the number of students in sixth grade through the quotient
[tex]\begin{gathered} \frac{180}{15}=number\text{ of students} \\ 12=numberofstudents \end{gathered}[/tex]1 pointGiven: AABCDEF. What is the missing congruent to prove that the triangles are congruent by AAS?EF10000DEABDF ACLE LBLF LBPrevious
Answers
From the problem we can conclude that the missing congruency is:
[tex]DF\cong AC[/tex]Brian is running a race. He runs for 17.5 miles at a speed of 7 miles per hour. For how many hours does he run?
Answers
17.5 miles at a speed of 7 miles per hour.
time = miles/ speed
time = 17.5 miles * 1hour/ 7 miles = 5/2 = 2.5 hours
_________________________________
For how many hours does he run?
He ran 2.5 hours
Help please please please help help Standard Deviation of Wild Cat
Answers
Solution
We are given the following data for various wild large cats
[tex]65,50,35,50,45,25,40,40,20[/tex]here, n = 9
Note: Standard deviation Formula
The x - bar denotes the mean
First, we find the mean
[tex]\begin{gathered} \bar{x}=\frac{65+50+35+50+45+25+40+40+20}{9} \\ \bar{x}=\frac{370}{9} \end{gathered}[/tex]Now we will do x - mean, we have the answers as follows
[tex]\frac{215}{9},\frac{80}{9},-\frac{55}{9},\frac{80}{9},\frac{35}{9},-\frac{145}{9},-\frac{10}{9},-\frac{10}{9},-\frac{190}{9}[/tex]We will find their squares
[tex]\begin{gathered} Standard\text{ }Deviation\sqrt{\frac{(\frac{215}{9})^2+2(\frac{80}{9})^2+(\frac{55}{9})^2+(\frac{35}{9})^2+(\frac{145}{9})^2+2(\frac{10}{9})^2+(\frac{190}{9})^2}{9}} \\ \\ Standard\text{ }Deviation= \\ \end{gathered}[/tex]elena wants to estimate 197.6÷5.48
Answers
The number 197.6 can be estimated to be somewhere around 200
The number 5.48 can be estimated to be somewhere around 5
Thus, the estimation problem becomes:
200 ÷ 5
This is easily found to be:
200 ÷ 5 = 40
The estimation is
40Three congruent circles touch one another as shown in the figure. The radius of each circleis 6 cm. Find the area of the unshaded region within the triangle ABC
Answers
Option A is correct
Area of the unshaded region = 18(2√3 - π)
Explanations:Note that triangle ABC is an equilateral triangle, therefore the area of triangle ABC will be found using the formula for the area of an equilateral triangle
[tex]\text{Area of triangle ABC = }\frac{\sqrt[]{3}}{4}a^2[/tex]where a represents each side of the triangle.
In triangle ABC , a = 6 + 6
a = 12 cm
[tex]\begin{gathered} \text{Area of triangle ABC = }\frac{\sqrt[]{3}}{4}(12^2) \\ \text{Area of triangle ABC = }\frac{144\sqrt[]{3}}{4} \\ \text{Area of triangle ABC = }36\sqrt[]{3} \end{gathered}[/tex]There are three sectors contained in the triangle, and each of them form an angle 60° with the center.
The radius, r = 6 cm
[tex]\begin{gathered} \text{Area of each sector = }\frac{\theta{}}{360}\times\pi r^2 \\ \text{Area of each sector = }\frac{60}{360}\times\pi\times6^2 \\ \text{Area of each sector = 6}\pi \end{gathered}[/tex]Area of the three sectors contained in the triangle = 3(6π)
Area of the three sectors contained in the triangle = 18π
Area of the unshaded region = (Area of the triangle ABC) - (Total Area of the sectors)
Area of the unshaded region = 36√3 - 18π
Area of the unshaded region = 18(2√3 - π)