Three 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)
Related Questions
Write the point-slope form of the equation of the line with slope -7/4 that passes through the point (-9, 2).
Answers
We have the slope
m= -7/4
and the point (x0,y0) = (-9,2).
The point-slope equation is given by
y-yo = m(x-x0)
y-2 = -7/4(x + 9)
Answer: the second option
A graph artist was painting a logo in the shape of a regular polygon. She measured one of the exterior angle to be 45 degrees. How many sides does this polygon has? What is the name of such polygon?
Answers
We are given a polygon
with exterior angle= 45
sides=?
since the polygon exterior angle is 45
the interior angle is
[tex]180-45=135[/tex]the only polygon with inner angles equals to 135 is a octagon
other way to find the octagon is as we know that all the exterior angles sum to 360
then
[tex]\frac{360}{45}[/tex][tex]\frac{360}{45}=8[/tex]-6=61–2 – h) + 3 + j; use h = 3, and j = 6
Answers
For this case we have the following expression given:
[tex]\frac{-6}{6(-2-h)}+3+j\text{ , h= }3\text{, j=6}[/tex]We can begin doing this:
[tex]\frac{-6}{-6(2+h)}+3+j[/tex]We can cancel -6 and we got:
[tex]\frac{1}{2+h}+3+j[/tex]Now we can replace h=3 and j= 6 and we got:
[tex]\frac{1}{2+3}+3+6=\text{ }\frac{1}{5}+9=\text{ }\frac{46}{5}[/tex]Type the expression that results from the following series of steps:
Start with
z
, times by 5, then subtract
t
.
Answers
Type the expression that results from the following series of steps:
Start with
z
, times by 5, then subtract
t
.
i need help please i need the domain and range using interval notation some one help me fast please
Answers
the domain tells that the how much the variable x is varying horizontally from one end to another end of the graph
in this graph, we can see that there are two points that are starting and endpoint
the starting of the graph is from x = -2 and it is ending at x = 4
but at x = 4 the graph is showing a void circle that means x is just less than 4 but it is not 4
so the domain is [-2, 4)
and the range represents y highest value and lowest value.
so the range of the graph is y = -3 to y = 3
at y = - 3 so we will this ) because -3 is not the graph, the value of y is just less than.
so the ranges is (-3, 3]
I need help with this practice problem *** I will send another picture that goes along with this problem, it is a graph it asks to use it to answer. It’s trig
Answers
Check the answer below, please.
1) Before plotting this, we need to remind ourselves the definition of Midline. Roughly saying midline is located halfway between the maximum and the minimum point in a sinusoidal graph.
2) So, bearing in mind the given function we can plot the following:
Note that the dashed line is the midline of that cosine function that is horizontally translated to the left pi/2 units.
Note that the midline marks halfway the maximum and the minimum point.
Thus this is the graph of the function
(3w)(w-2)=(w-1)(w) find the value of w
Answers
(3w)(w-2)=(w-1)(w) find the value of w
[tex]\begin{gathered} \mleft(3w\mright)\mleft(w-2\mright)=\mleft(w-1\mright)\mleft(w\mright) \\ \text{Apply distributive property} \\ 3w^2-6w=w^2-w \\ 3w^2-w2=6w-w \\ 2w^2=5w \\ 2w=5 \\ w=\frac{5}{2} \\ w=2.5 \end{gathered}[/tex]Zeem wants to buy a new cell phone. She has researched and found th following four offers: AT&T $389.99 with a $15.00 discount T-Mobile $400.99 with a $10 mail-in rebate Sprint $379.00 with a 10% discount Straight Talk $400 with a 20% discount Which offer will cost Ms. Zeem the least to buy her phone?
Answers
In this case, we'll have to carry out several steps to find the solution.
Step 01:
Data
AT&T $389.99 ===> $15.00 discount
T-Mobile $400.99 ====> $10 mail-in rebate
Sprint $379.00 ===> 10% discount
Straight Talk $400 ===> 20% discount
Step 02:
Cost
AT&T
389.99 - 389.99*0.15 = 331.49
T-Mobile
400.99 - 10 = 390.99
Sprint
379.00 - 379.00*0.1 = 341.1
Straight Talk
400 - 400*0.2 = 320
The answer is:
Straight Talk it is the lowest cost = $320
Find the equation of the long with slope -2 and that contains the point (-8,-2). Write the equation in the form y=mx+b and identify m and bm=b=
Answers
Using the equation y=mx + b with m=-2 and the point (-8,-2) to find the y-intercept (b), we have:
-2=-2(-8) + b (Replacing the values)
-2= 16 + b (Multiplying)
-18 = b (Subtracting 16 from both sides of the equation)
The answers are:
Equation of the line: y = -2x -18
m= -2
b= -18
which expression represents 87 less than the product of 12 and 15 A 87-(12×15)B 87×(15-12)C (12×15)-87
Answers
The expression that represents 87 less than the product of 12 and 15 is C.
Find the vertex and write the quadratic function in vertex form.f(x)=x^2−6x+25
Answers
A quadratic equation in its standard formula y = ax² + bx + c, can also be written in the vertex form: y = a(x - h)² + k where the point (h, k) is the vertex of the parabola.
Then, to solve this question, follow the steps below.
Step 01: Find x-vertex.
The x-vertex (h) can be found using the equation:
[tex]h=\frac{-b}{2a}[/tex]In this equation,
b = -6
a = 1
Then,
[tex]\begin{gathered} h=-\frac{(-6)}{2\cdot1} \\ h=\frac{6}{2} \\ h=3 \end{gathered}[/tex]Step 02: Substitute x by 3 in the standard form to find y-vertex (k):
[tex]\begin{gathered} y=x^2-6x+25 \\ y=3^2-6\cdot3+25 \\ y=9-18+25 \\ y=-9+25 \\ y=16 \end{gathered}[/tex]So, k = 16.
Step 03: Substitute the values in the vertex form.
a = 1
h = 3
k = 16
[tex]\begin{gathered} y=a\cdot(x-h)^2+k \\ y=1\cdot(x-3)^2+16 \\ y=(x-3)^2+16 \end{gathered}[/tex]Answer:
[tex]y=(x-3)^2+16[/tex]Resolver los siguientes sistemas de ecuaciones, empleando el método analítico algebraico que gustes en cada caso (sustitución, igualación, reducción, determinantes, etc.), solo se te solicita que emplees al menos dos métodos diferentes, a lo largo de este primer ejercicio
3x+2y=1
-x+3y=7
Answers
The solution to the system of equations (x,y) are (-1, 2).
System of EquationA system of equations is a group of two or more equations with the same variables.
A solution to a system of equations is the values of the variables that make all of the equations in the system true. A solution to a system is also an intersection point of the graphs of the equations in the system.
To solve this problem, we need to find x and y which will give us two solutions to the problem.
3x + 2y = 1 ... equation (i)-x + 3y = 7 ... equation (ii)From equ(ii), make x the subject of formulax = 3y - 7(iii)Substitute equ(iii) into equ(i)3(3y - 7) + 2y = 1Solve the equation above9y - 21 + 2y = 111y - 21 = 111y = 22y = 2Put y = 2 into equ(i) (ii)From equation (i);3x + 2y = 13x + 2(2) = 13x + 4 = 13x = -3 x = -1The value of x and y are -1 and 2 respectively.
Learn more on system of equation here: https://brainly.com/question/13729904
#SPJ1
Find an equation of the ellipse that has center (1.-4), a major axis of length 12, and endpoint of minor axis (4.-4)
Answers
Solution:
Given:
Ellipse with the following properties;
[tex]\begin{gathered} \text{centre}=(1,-4) \\ \text{length of major axis= 12} \\ \text{End point of minor axis = (4,-4)} \end{gathered}[/tex]The equation of an ellipse is given by;
[tex]\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1[/tex]where,
[tex]\begin{gathered} (h,k)\text{ is the centre} \\ (h,k)=(1,-4) \\ h=1 \\ k=-4 \end{gathered}[/tex]Substituting these values into the equation of an ellipse,
[tex]\begin{gathered} \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1 \\ \frac{(x-1)^2}{a^2}+\frac{(y-(-4))^2}{b^2}=1 \\ \frac{(x-1)^2}{a^2}+\frac{(y+4)^2}{b^2}=1 \end{gathered}[/tex]The length of the semi-minor axis (a) is the difference between the x-values of the centre and the endpoint of the minor axis.
Hence,
[tex]\begin{gathered} a=4-1 \\ a=3 \\ a^2=3^2 \\ a^2=9 \end{gathered}[/tex]To get the length of the semi-major axis (b),
[tex]\begin{gathered} \text{length of the major axis=2b} \\ 12=2b \\ b=\frac{12}{2} \\ b=6 \\ b^2=6^2 \\ b^2=36 \end{gathered}[/tex]Substituting these into the equation of the ellipse,
[tex]\begin{gathered} \frac{(x-1)^2}{a^2}+\frac{(y+4)^2}{b^2}=1 \\ \frac{(x-1)^2}{9^{}}+\frac{(y+4)^2}{36^{}}=1 \end{gathered}[/tex]Therefore, the equation of the ellipse is;
[tex]\frac{(x-1)^2}{9^{}}+\frac{(y+4)^2}{36^{}}=1[/tex]
4711 13A BC DGiven this number line, CD = [?]
Answers
To obtain the measure of CD, find the absolute value of the difference of the coordinates.
[tex]\begin{gathered} CD=|13-11| \\ CD=|2| \\ CD=2 \end{gathered}[/tex]How do you do this? I know how to figure out C but im not sure about B?
Answers
Answer: 50
Step-by-step explanation: i Think..
⇒To figure out b you have already been given an equation :[tex]a^{2} +b^{2} =c^{2}[/tex]
derived by Pythagoras
⇒n order to find the value of b the then in the given equation make b the subject of the formula:
[tex]b^{2} =c^{2} -a^{2} \\b=\sqrt{c^{2}-b^{2} }[/tex]
where your c us the hypotenuse equal to 6
where b is the adjacent side
where a is the opposite side equal 4
[tex]b=\sqrt{(6)^{2} -(4)^{2} } \\b=\sqrt{36-16} \\b=\sqrt{20} \\b=2\sqrt{5}[/tex]
GOODLUCK!!
what are three equivalent Ratios for 8/10
Answers
Answer: 16/20, 24/30, and 32/40
We are given a ratio of 8 to 10
[tex]\begin{gathered} \frac{8}{10} \\ \text{ To find the equivalent ratios, the numerator must be a multiple of 8 and the denominator must be a multiple of 10} \\ \text{Multiple of 8: 8 x 1, 8 x 2, 8 x 3, }8\text{ x 4} \\ \text{Multiple of 8: 8, 16, 24, 32} \\ \text{Multiple of 10: 10 x 1, 10 x 2, 10 x 3, 10 x 4} \\ \text{Multiple of 10: 10, 20, 30, 40} \\ \text{For 8 : 10} \\ \frac{8}{10\text{ }}\text{ = }\frac{4}{5} \\ \frac{16}{20}\text{ = }\frac{4}{5} \\ \frac{24}{30}\text{ = }\frac{4}{5} \\ \frac{32}{40}\text{ = }\frac{4}{5} \end{gathered}[/tex]Therefore, the equivalent ratios for 8/10 are 16/20, 24/30, and 32/40
Translate to an algebraic expression, but do not simplify.subtract 13 from -25
Answers
Given:
Subtract 13 from -25.
To find:
a) The algebraic expression.
b) Simplify
Explanation:
a)
The algebraic expression of the given statement is,
[tex]-25-13[/tex]b)
On simplification we get,
[tex]-25-13=-38[/tex]Therefore, the simplified algebraic expression is,
[tex]-38[/tex]Final answer:
The algebraic expression is,
[tex]-25-13[/tex]The simplified algebraic expression is,
[tex]-38[/tex]{(-8,-10), (-2,0), (1,10), (4,0)}Domain=Range=Function?
Answers
The domain is all the elements of the first component, the range is all the elements of the second component and it is a function if every first component is only related to only one second component.
Therefore:
Domain = { -8, -2, 1 , 4}
Range = { -10, 0, 10}
Function = yes!
Answer: Domain = { -8, -2, 1 , 4}
Range = { -10, 0, 10}
Function = yes!
what is equivalent to this product below?[tex] \sqrt{9} \times \sqrt{5} \times \sqrt{2} [/tex]
Answers
We need to simplify the product:
[tex]\sqrt{9}\times\sqrt{5}\operatorname{\times}\sqrt{2}[/tex]Notice that:
[tex]\begin{gathered} \sqrt{9}=3 \\ \\ \text{since }3²=9 \end{gathered}[/tex]Also, we have:
[tex]\sqrt{5}\operatorname{\times}\sqrt{2}=\sqrt{5\times2}=\sqrt{10}[/tex]Therefore, we have:
[tex]\sqrt{9}\times\sqrt{5}\times\sqrt{2}=3\sqrt{10}[/tex]Answer: A. 3√10
how do I find square footage. my question is Pam needed to give her gardener the approximate square footage of her backyard. With each step being about 2 feet, Pam steps it off and counts 20 steps long and 40 steps wide. About how many square feet is the backyard?
Answers
Explanation:
Since each of her steps is equivalent to 2 feet
=> 1 step = 2 ft
==> 20 steps = 2 * 20 = 40 feet
==> 40 steps = 2 * 40 = 80 feet
The square feet of the backyard is equal to width * height
[tex]40\cdot80=3,200ft^2[/tex]I have a class to finish it ends in two days
Answers
The given equation is:
[tex]x^2+y^2-x-2y-\frac{11}{4}=\text{ 0}[/tex]The standard equation of a circle is given as:
(x - a)² + (y - b)² = r²
where (a, b) is the center
r is the radius
Express the given equation in form of the standard equation
Collect like terms
[tex]x^2-x+y^2-2y\text{ = }\frac{11}{4}[/tex]Add the squares of the half of the coefficients of x and y to both sides of the equation
[tex]\begin{gathered} x^2-x+(\frac{-1}{2})^2+y^2-2y+(-1)^2=\frac{11}{4}+(\frac{-1}{2})^2+(-1)^2_{} \\ x^2-x+(\frac{1}{2})^2+y^2-2y+1^2=\frac{11+1+4}{4} \\ (x-\frac{1}{2})^2+(y-1)^2=\frac{16}{4} \\ (x-\frac{1}{2})^2+(y-1)^2=\text{ 4} \\ (x-\frac{1}{2})^2+(y-1)^2=2^2 \end{gathered}[/tex]Compare the resulting equation with (x - a)² + (y - b)² = r²
[tex]\begin{gathered} \text{The center (a, b) = (}\frac{1}{2},\text{ 1)} \\ \text{The radius, r = 2} \end{gathered}[/tex]A professor went to a website for rating professors and looked up the quality rating and also the "easiness" of the six full-time professors in onedepartment. The ratings are 1 (lowest quality) to 5 (highest quality) and 1 (hardest) to 5 (easiest). The numbers given are averages for eachprofessor. Assume the trend is linear, find the correlation, and comment on what it means.
Answers
We are asked to determine the correlation factor "r" of the given table. To do that we will first label the column for "Quality" as "x" and the column for "Easiness" as "y". Like this:
Now, we create another column with the product of "x" and "y". Like this:
Now, we will add another column with the squares of the values of "x". Like this:
Now, we add another column with the squares of the values of "y":
Now, we sum the values on each of the columns:
Now, to get the correlation factor we use the following formula:
[tex]r=\frac{n\Sigma xy-\Sigma x\Sigma y}{\sqrt{(n\Sigma x^2-(\Sigma x)^2)(n\Sigma y^2-(\Sigma y)^2)}}[/tex]Where:
[tex]\begin{gathered} \Sigma xy=\text{ sum of the column of xy} \\ \Sigma x=\text{ sum of the column x} \\ \Sigma y=\text{ sum of the column y} \\ \Sigma x^2=\text{ sum of the column x\textasciicircum2} \\ \Sigma y^2=\text{ sum of the column y\textasciicircum2} \\ n=\text{ number of rows} \end{gathered}[/tex]Now we substitute the values, we get:
[tex]r=\frac{\left(6)(70.56)-(25.2)(16.4\right)}{\sqrt{((6)(107.12)-(25.2)^2)((6)(47.82)-(16.4)^2)}}[/tex]Solving the operations:
[tex]r=0.858[/tex]Therefore, the correlation factor is 0.858. If the correlation factor approaches the values of +1, this means that there is a strong linear correlation between the variables "x" and "y" and this correlation tends to be with a positive slope.
Find the volume of the figures.A. 907.5 cmB. 126ncmC. 605 cmD. 55 cm
Answers
We have the next formula to calculate the volume
[tex]V=\frac{1}{3}\cdot a^2\cdot h[/tex]where a is the length of the side in the base and h is the height
a=11cm
h=15cm
we substitue the values in the formula
[tex]V=\frac{1}{3}\cdot11^2\cdot15=605\operatorname{cm}^3[/tex]A) Find the circumference of a circle whose radius is 4 inches.Round to the nearest tenth.B) Find the length of AB, if m
Answers
Given:
• Radius of the circle, r = 4 inches
,• Central angle, m∠ACB = 60 degrees
Let's solve for the following:
• (A). Find the circumference of a circle whose radius is 4 inches.
To find the circumference, apply the formula:
[tex]C=2\pi r[/tex]Where:
C is the circumference.
r is the radius = 4 inches.
Plug in 4 for r and solve for r:
[tex]\begin{gathered} C=2\pi *4 \\ \\ C=25.1\text{ inches} \end{gathered}[/tex]Therefore, the circumference of the circle is 25.1 inches.
• (B). Let's find the length of the arc AB.
To find the length of the arc AB, apply the formula:
[tex]\begin{gathered} L=2\pi r*\frac{\theta}{360} \\ \\ L=C*\frac{\theta}{360} \end{gathered}[/tex]Where:
θ is the central angle = 60 degrees.
C is the circumference.
Thus, we have:
[tex]\begin{gathered} L=25.1*\frac{60}{360} \\ \\ L=25.1*\frac{1}{6} \\ \\ L=4.18\approx4.2\text{ inches} \end{gathered}[/tex]Therefore, the length of the arc is 4.2 inches.
ANSWER:
• (A). 25.1 inches
• (B). 4.2 inches
Which equation represents a line which is parallel to the line y = 3x - 8x + y = 18x - 3y = -123x + y = 6y - 3x = 7
Answers
Parallel equations have the same slope. First, let's identify the slope of the given equation, which is in the slope-intercept form:
[tex]y=mx+b[/tex]Where m is the slope and b is the y-intercept.
Comparing the given equation with this form, we have m = 3, so the slope is 3.
Now, let's identify the slope of each option:
[tex]\begin{gathered} x+y=18\\ \\ y=-x+18\\ \\ m=-1 \end{gathered}[/tex][tex]\begin{gathered} x-3y=-12\\ \\ 3y=x+12\\ \\ y=\frac{1}{3}x+4\\ \\ m=\frac{1}{3} \end{gathered}[/tex][tex]\begin{gathered} 3x+y=6\\ \\ y=-3x+6\\ \\ m=-3 \end{gathered}[/tex][tex]\begin{gathered} y-3x=7\\ \\ y=3x+7\\ \\ m=3 \end{gathered}[/tex]Therefore the correct option is the fourth one: y - 3x = 7.
A Ferris Wheel has a center of 85 feet off the ground and the highest car sits 166 feet off the ground.
Answers
As the highest car sits 166 feet off the ground. the radius of the circle is equal to the difference between that heigth and the center of the ferris:
[tex]\begin{gathered} r=166ft-85ft \\ \\ r=81ft \end{gathered}[/tex]The radius is 81 feetThe lowest a cat will sit is the difference between the center of the circle and the radius:
[tex]85ft-81ft=4ft[/tex]The lowest a car will sit on the Ferris wheel is 4ftYou are choosing two cards, replacing the first card in the deck after it has been drawn. What is the probability you choose two consecutive fours?
Answers
With replacement:
[tex]P=\frac{4}{52}\cdot\frac{4}{52}=\frac{16}{2704}=\frac{1}{169}[/tex]there are four 4s in the 52-card deck -> 4/52
Since you return that card into the deck, the probability of getting two consecutive 4s will be (4/52)*(4/52)
For the following equation determine the value of the missing entries reduce all fractions to lowest term note each column in the table represents an ordered pair. if multiple solutions exist you only need to identify one
Answers
Given:
[tex]8x-4y=18[/tex]To complete the table, let's substitute the values that are known and find the unknown.
a) y = 0
[tex]\begin{gathered} 8x-4*0=18 \\ 8x=18 \\ Dividing\text{ }both\text{ sides }by\text{ 8:} \\ \frac{8x}{8}=\frac{18}{8} \\ x=\frac{18}{8} \\ Dividing\text{ }the\text{ }numerator\text{ and }the\text{ }denominator\text{ }by\text{ 2:} \\ x=\frac{\frac{18}{2}}{\frac{8}{2}} \\ x=\frac{9}{4} \end{gathered}[/tex]The first point is (9/4, 0).
b) x = 0
[tex]\begin{gathered} 8*0-4y=18 \\ -4y=18 \\ Dividing\text{ }the\text{ }sides\text{ }by\text{ -4:} \\ -\frac{4y}{-4}=\frac{18}{-4} \\ y=-\frac{18}{4} \\ Dividing\text{ }by\text{ }2: \\ y=-\frac{\frac{18}{2}}{\frac{4}{2}} \\ y=-\frac{9}{2} \end{gathered}[/tex]The first point is (0, -9/2).
c) x = 1
[tex]\begin{gathered} 8*1-4y=18 \\ 8-4y=18 \\ Subtracting\text{ }8\text{ }from\text{ both }sides: \\ 8-4y-8=18-8 \\ -4y=10 \\ Dividing\text{ }the\text{ }sides\text{ }by\text{ }-4: \\ \frac{-4y}{-4}=\frac{10}{-4} \\ y=-\frac{10}{4} \\ Dividing\text{ }the\text{ }sides\text{ }by\text{ 2:} \\ y=-\frac{\frac{10}{2}}{\frac{4}{2}} \\ y=-\frac{5}{2} \end{gathered}[/tex]The third point is (1, -5/2).
d) y = 3
[tex]\begin{gathered} 8x-4*3=18 \\ 8x-12=18 \\ Adding\text{ }12\text{ }to\text{ }both\text{ }sides: \\ 8x-12+12=18+12 \\ 8x=30 \\ Divind\text{ }by\text{ 8:} \\ \frac{8x}{8}=\frac{30}{8} \\ x=\frac{30}{8} \\ Divind\text{ }by\text{ 2: } \\ x=\frac{\frac{30}{2}}{\frac{8}{2}} \\ x=\frac{15}{4} \end{gathered}[/tex]The fourth point is (15/4, 3).
Answer:
x 9/4 0 1 15/4
y 0 -9/2 -5/2 3
Need answers to check my work :) quick and simple.*just a math practice
Answers
7.
Alternative interior angles
8.
Alternative exterior angles
9.
Theorem vertical angles
10.
Theorem supplementary angles
11.
Alternative exterior angles
12.
Theroem congruent angles
I only need the correct answer. This is part one 1 out of 3.
Answers
To find the range, we subtract the lowest value from the greatest value, ignoring the others.
So, in this case, we have:
• Lowest value: 20.3
,• Greatest value: 110.4
[tex]\text{Range = }110.4-20.3=90.1[/tex]Therefore, the range is 90.1
Part 2
The formula to find the variance is
[tex]\begin{gathered} \sigma^2=\frac{\sum ^{}_{}(x-\mu)^2}{n} \\ \text{ Where} \\ x=\text{ data values} \\ \mu=\text{ mean} \\ n=\text{ number of data points} \end{gathered}[/tex]The formula to find the mean is
[tex]\mu=\frac{\sum ^{}_{}x}{n}[/tex]So, as you can see, we first find the mean, and with this value, we find the variance of the data set.
• Mean
[tex]\begin{gathered} \mu=\frac{20.3+33.5+21.8+58.2+23.2+110.4+30.9+24.4+74.6+60.4+40.8}{11} \\ \mu=\frac{498.5}{11} \\ \mu=45.32 \end{gathered}[/tex]• Variance
[tex]\begin{gathered} \sigma=\frac{(20.3-45.32)^2+(33.5-45.32)^2+(21.8-45.32)^2+(58.2-45.32)^2+(23.2-45.32)^2+(110.4-45.32)^2+(30.9-45.32)^2+(24.4-45.32)^2+(74.6-45.32)^2+(60.4-45.32)^2+(40.8-45.32)^2}{11} \\ \sigma=\frac{(-25.02)^2+(-11.82)^2+(-23.52)^2+(12.88)^2+(-22.12)^2+(65.08)^2+(-12.42)^2+(-20.92)^2+(29.28)^2+(15.08)^2+(-4.52)^2}{11} \\ \sigma=\frac{625.91+139.67+553.10+165.94+489.21+4325.64+207.88+437.57+857.42+227.46+20.41}{11} \\ \sigma=\frac{7960.24}{11} \\ $$\boldsymbol{\sigma=723.66}$$ \end{gathered}[/tex]Therefore, the variance is 723.66.