Answers
1) Problem 17.
(i) Determine whether T is continuous at 6061.
For that you have to compute the value of T at 6061 and the lateral limits of T when x approaches 6061.
a) T(x) = 0.10x if 0 < x ≤ 6061
T (6061) = 0.10(6061) = 606.1
b) limit of Tx when x → 6061.
By the left the limit is the same value of T(x) calculated above.
By the right the limit is calculated using the definition of the function for the next stage: T(x) = 606.10 + 0.18 (x - 6061)
⇒ Limit of T(x) when x → 6061 from the right = 606.10 + 0.18 (6061 - 6061) = 606.10
Since both limits and the value of the function are the same, T is continuous at 6061.
(ii) Determine whether T is continuous at 32,473.
Same procedure.
a) Value at 32,473
T(32,473) = 606.10 + 0.18 (32,473 - 6061) = 5,360.26
b) Limit of T(x) when x → 32,473 from the right
Limit = 5360.26 + 0.26(x - 32,473) = 5360.26
Again, since the two limits and the value of the function have the same value the function is continuos at the x = 32,473.
(iii) If T had discontinuities, a tax payer that earns an amount very close to the discontinuity can easily approach its incomes to take andvantage of the part that results in lower tax.
2) Problem 18.
a) Statement Sk
You just need to replace n for k:
Sk = 1 + 4 + 7 + ... (3k - 2) = k(3k - 1) / 2
b) Statement S (k+1)
Replace
S(k+1) = 1 + 4 + 7 + ... (3k - 2) + [ 3 (k + 1) - 2 ] = (k+1) [ 3(k+1) - 1] / 2
Simplification:
1 + 4 + 7 + ... + 3k - 2+ 3k + 3 - 2] = (k + 1) (3k + 3 - 1)/2
k(3k - 1)/ 2 + (3k + 1) = (k + 1)(3k+2) / 2
Do the operations on the left side and you will find it can be simplified to k ( 3k +1) (3 k + 2) / 2.
With that you find that the left side equals the right side which is a proof of the validity of the statement by induction.
Please, see the detailed solution in the attached files.
Thanks.
Related Questions
Acapulco, Mexico and Hyderabad, India both lie at 17° north latitude, and lie very nearly halfway around the world from each other in an east-west direction. The radius of Earth at a latitude of 17° is about 3790 miles. Suppose that you could fly from Acapulco directly west to Hyderabad or fly directly north to Hyderabad. Which way would be shorter, and by how much? Use 3960 miles for Earth’s radius. (Hint: To fly directly north, you would go from 17° north latitude to 90° north latitude, and then back down to 17° north latitude.)
Answers
[tex]L=2 \pi R \frac{C}{360} [/tex]
Where:
L= arc length (in this problem it is disance we need to travel)
R = radius of circle (in this problem it is equal to a radius of Earth)
C = angle we need to pass
We are told that two cities lie halfway around the world. If we fly to west this means angle is 180°.
This gives an arc length of:
[tex]L=2* \pi *3960* \frac{180}{360} \\ \\ L=12440.71 miles[/tex]
If we want to fly to north we need to go to 90° northern latidtude and then back to 17° latitude. This means angle is:
C=2*(90-17)=2*73°=146°
This gives an arc length of:
[tex]L=2* \pi *3960* \frac{146}{360} \\ \\ L=10090.8 miles[/tex]
We can see that flying north is shorter. It is shorter by:
12440.71 miles - 10090.8 miles = 2349.91 miles
Choose as many answers as apply.
An employee can legitimately claim the following as exemptions:
A. family pet
B. self
C. dependents
D. neighbor's cousin Al
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Answer:
An employee can legitimately claim the following as exemptions:
B. Self
C. Dependents
Step-by-step explanation:
We know that an employee can claim the health and medical insurance ( which is also known as Medicare and Social security deductions) for himself or for himself, his/ her spouse and his family members or dependents.
Hence,
An employee can legitimately claim the following as exemptions:
B. Self
C. dependents.
Hence, option B and C are the correct options.
What is the volume of the cone to the nearest tenth?
Height: 14 cm
Radius: 5 cm
A. 549.8 cm3
B. 73.3 cm3
C. 366.5 cm3
D. 1,026.3 cm3
Answers
=1099/3
=366.5cmcube
A class has 24 boys and 16 girls. on a test the class mean was 75. the mean of the girls' score was 72. what was the mean of the boys' scores?
Answers
40*75=3000
16*72=1152
3000-1152=1848
1848/24=77
The mean of the boys' scores was 77.
What is arithmetic mean?The arithmetic mean is defined as the ratio of the sum of observations to the total number of observations. It can be referred to as the average of a specific set of data or the arithmetic mean. One of the basic methods used to acquire a result, the term "Mean" is commonly used in the field of statistics.
We have been given that the class has 24 boys and 16 girls.
And the class mean was 75
The total number of students in the class was:
24 + 16 = 40
Total scored by 40 students = 40 × 75 = 3000
Number of girls = 16 and their mean = 72
Total scored by girls = 16 × 72 = 1152
The difference between the scores will give the total score of boys:
⇒ 3000 - 1152 = 1848
Number of boys = 24
Their mean will be 1848 / 24
So the mean of the boys' scores = 77
Therefore, the mean of the boys' scores was 77.
Learn more about Arithmetic means here:
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Please Help!!!
Let x2−12x=61 . What values make an equivalent number sentence after completing the square?
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The given equation is [tex] x^2-12x=61 [/tex]. Add 36 to both the sides of the equation, then
[tex] x^2-12x+36=61+36 [/tex].
The equation becomes,
[tex] x^2-12x+6^2=97\\
(x-6)^2=97 [/tex]
EASY 14 POINTS, PLEASE HELP ASAP.
Go into a dark room with a flashlight and face the beam toward a wall. Change the flashlight's distance from the wall as well as the way you hold the flashlight. Which conic sections can you form? Explain.
Describe how you formed different conic sections with your flashlight. Take and upload digital pictures as needed. Look at other students' descriptions and pictures and tell whether you agree with them.
What other simple tasks can you do in your home to form any of the conic sections? Again, describe these and/or take pictures to share with other students.
Answers
Well really it would have to depend on the flashlight. Is it an led bulb or a regular bulb?
A new fence is constructed on three sides of a housing block with area 240 m2. the fourth side facing the road is left open. if 52 m of fencing is used, what are the dimensions of the block?
Answers
thus
2x+y=52
hence
y=52-2x
area is given by:
area=xy
x(52-2x)=240
re-writing the above we get
-2x^2+52x-240=0
x^2-26x+120=0
factoring the above we get:
(x-20)(x-6)=0
thus
x=20 or x=6
if x=20, then y=52-40=12
if x=6, then y=52-12=40
To find the housing block dimensions, we set up a system of equations based on the total fencing and area given, and then solve for the length and width.
Explanation:The student has asked about finding the dimensions of a housing block where a new fence is constructed on three sides and the total area of the block is 240 m2. Given that 52 m of fencing is used, to determine the dimensions, we need to solve a system of equations derived from the perimeter and area provided.
Let's assume the length of the block is L meters, and the width is W meters. The area of the block (A) can thus be defined as A = L × W. Given that A = 240 m2, we have L × W = 240. Since the fence covers three sides, the total length of fencing used would be L + 2W, which is equal to 52 meters. Therefore, we have L + 2W = 52 meters.
To solve for L and W, we set up the system of equations:
L × W = 240L + 2W = 52From the second equation, we can express L as L = 52 - 2W and then substitute into the first equation to find W. Finally, we can determine the value for L using the found value of W.
Let 40 x 55 = n. Which of the choices shown is also equal to n?
A) (550 x 40)
B) (4 x 50) + (40 x 5)
C) (50 x 40) + (50 x 40)
D) (10 x 55) + (10 x 55) + (10 x 55) + (10 x 55)
Answers
use distributive rule to factor out 55:
55 (10+10+10+10) = 55×40
Evaluate 8 x 6 + (12- 4) divided by 2
Answers
12 - 4 = 8
8 x 6 = 48
48 + 8 = 56
56/2 = 28
Therefore, the solution of given equation is [tex]28.[/tex]
What is the simplification?
Simplification is reducing the expression/fraction/problem in a simpler form. It makes the problem easy with calculations and solving.
Here given equation is
[tex]8[/tex] x [tex]6 + (12- 4)[/tex] which is divisible by [tex]2[/tex].
So,
[tex]\frac{8(6)+(12-4)}{2}\\\\=\frac{48+8}{2}\\\\=\frac{56}{2}\\\\=28[/tex]
Hence, the solution of given equation is [tex]28.[/tex]
To know more about the simplification
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For this tangent line approximation question, my answer is D. 4. Is my work correct?
f(x) = f(a) + (x - a)f'(a)
a = -0.5
x = 0
f(0) = 2 + 0.5(4) = 4
Answers
If you wanted, you can simplify the linear approximation L(x)
L(x) = f(a) + (x-a)*f'(a)
L(x) = f(-0.5) + (x-(-0.5))*f'(-0.5)
L(x) = 2 + (x+0.5)*4
L(x) = 2+4x+2
L(x) = 4x+4
Then plug in x = 0
L(x) = 4x+4
L(0) = 4(0)+4
L(0) = 4
But your method is a shorter route.
Find the exact value of each expression . Do not use a calculator .
8. cos^{-1} (0)
9.sin^{-1} 1
10.tan^{-1} (\frac{\sqrt{3}}{3} )
I need help !!
Answers
8. [tex]\cos^{-1}(0)= 90^{\circ}=\dfrac{\pi}{2}[/tex]
9. [tex]\sin^{-1}(1)= 90^{\circ}=\dfrac{\pi}{2}[/tex]
10. [tex]\tan^{-1}\left( \dfrac{\sqrt{3}}{3} \right)=30^{\circ}=\dfrac{\pi}{6}[/tex]
The exact values of the given expressions are 90 degrees or π/2 radians, 90 degrees or π/2 radians, and 60 degrees or π/3 radians respectively.
Explanation:Question 8:
The inverse cosine (cos^{-1}) of 0 is the angle whose cosine is 0. In other words, cos^{-1}(0) will give you the angle whose cosine is 0. The only angle that satisfies this condition is 90 degrees or π/2 radians.
Question 9:
The inverse sine (sin^{-1}) of 1 is the angle whose sine is 1. In other words, sin^{-1}(1) will give you the angle whose sine is 1. The only angle that satisfies this condition is 90 degrees or π/2 radians.
Question 10:
The inverse tangent (tan^{-1}) of (√3/3) is the angle whose tangent is (√3/3). In other words, tan^{-1}(√3/3) will give you the angle whose tangent is (√3/3). The angle that satisfies this condition is 60 degrees or π/3 radians.
You and a friend go to a mexican restaurant. you order 2 tacos and 3 enchiladas and your friend orders 3 tacos and 5 enchiladas. one bill was $7.80 plus tax and the other bill was $12.70 plus tax. how much was each taco and each enchilada?
Answers
We have then:
x: cost of tacos
y: cost of enchiladas.
We write the system of equations:
2x + 3y = 7.80
3x + 5y = 12.70
Solving the system we have:
x = 0.9 $
y = 2 $
Answer:
each taco and each enchilada was:
x = 0.9 $
y = 2 $
To find the cost of each taco and enchilada, set up a system of equations based on the given bills and solve for the variables T (taco) and E (enchilada). The solution reveals that each taco costs $0.90 and each enchilada costs $2.00.
To solve the problem of determining the individual cost of a taco and an enchilada based on the two given bills, we can set up a system of linear equations. Let's denote the price of a taco as 'T' and the price of an enchilada as 'E'.
The first equation represents your order: 2T + 3E = $7.80.
The second equation represents your friend's order: 3T + 5E = $12.70.
To solve the system, you can use either the substitution or elimination method. For this example, we'll use the elimination method:
Multiply the first equation by 3 and the second by 2, which gives us:6T + 9E = $23.40
6T + 10E = $25.40Subtract the first new equation from the second new equation to eliminate 'T' and find the value of 'E'. This gives us E = $2.00.Now, plug the value of 'E' into the first original equation:
2T + 3(2.00) = $7.80
2T + $6.00 = $7.80
2T = $1.80
T = $0.90
So, each taco cost $0.90 and each enchilada cost $2.00.
For groups of students are doing a probability experiment with a standard number cube to see how many times they roll of four. The theoretical probability of rolling in for is one out of six or approximately 0.17 and one group came close to this possibility with an experimental probability of 0.175. This most likely came from which group
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Anyone know this geometry question? 10 min left! Will give brainiest
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The answer is B. m< 3 would need to double.
Hope this helps =)
Please help asap, 70 points, THREE questions (:
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[tex]ax^2+bx+c=0\\\\\Delta=b^2-4ac\\\\if\ \Delta > 0\ then\ the\ equation\ has\ 2\ solutions\\if\ \Delta=0\ then\ the\ equation\ has\ 1\ solution\\if\ \Delta < 0\ then\ the\ equation\ has\ no\ solution[/tex]
[tex]x^2+5x+7=0\\a=1;\ b=5;\ c=7\\\\Delta=5^2-4\cdot1\cdot7=25-28=-3 < 0\\\\\boxed{Answer:\ 0}[/tex]
[tex]5y^2-8y=2\ \ \ |-2\\\\5y^2-8y-2=0\\\\a=5;\ b=-8;\ c=-2\\\\b^2-4ac=(-8)^2-4\cdot5\cdot(-2)=64+40=104\\\\\sqrt{b^2-4ac}=\sqrt{104}\approx10.198\\\\y_1=\dfrac{-b-\sqrt{b^2-4ac}}{2a};\ y_2=\dfrac{-b+\sqrt{b^2-4ac}}{2a}\\\\y_1=\dfrac{-(-8)-10.198}{2\cdot5}\approx-0.22\\\\y_2=\dfrac{-(-8)+10.198}{2\cdot5}\approx1.82\\\\\boxed{Answer:\ 1.82,\ -0.22}[/tex]
Answer:
the answer for number 1 is 1.82
Step-by-step explanation:
Simplify 72x6 y7 z .
Answers
if it takes 18 gallons of gas to go 400 miles how many gallons would it take to go 300 miles
A. 5.5 gallons
B. 13.5 gallons
C. 16.6 gallons
D. 22.2 gallons
E. 24 gallons
Answers
We will use the ratio and proportion method ( an equation formed with two ratios that are equal) to solve the question “if it takes 18 gallons of gas to go 400 miles, how many gallons would it take to go 300 miles?”
Let x= the number of gallons for the gas to go 300 miles
18:400=X:300
400x=300*18
400x=5400
X=5400/400
X=13.5 so the answer is B.
Answer:
B
Step-by-step explanation:
its 13.5 gallons
The depth of a lake, d, varies directly with r, the amount of rainfall last month. if k is the constant of variation, write an equation to represent the situation.
Answers
d = kr
Explanation:
We know that the depth (d) varies directly with the amount of rain (r).
This means that as the amount of rain increases, the depth increases and vice-versa.
The depth would depend on r multiplied by a constant of variation (k)
Translating this into equation, we would get the following:
d = kr
Hope this helps :)
A.70
B.40
C.140
D.20
Answers
How to find amplitude and period from equation?
Answers
[tex]y = a \sin(bx) + c[/tex]
a is the Amplitude
and period is:
[tex] \frac{360}{b} [/tex]
If your graph is a TAN graph :
[tex]y = a \tan(bx) + c[/tex]
A TAN graph has no amplitude as it has no Maximum or Minimum value.
To find period :
[tex] \frac{180}{b} [/tex]
Final answer:
The amplitude of a wave is the coefficient A in the equation y(x, t) = A sin(kx - ωt + φ), and the period T can be calculated using T = 2π/ω, where ω is the angular frequency. Additional properties like wavelength and speed can be found using related equations involving the wave number and frequency.
Explanation:
To find the amplitude and period from the equation of a wave, look at the standard form of a sinusoidal wave function, which is generally y(x, t) = A sin(kx - ωt + φ). Here, the amplitude can be directly read from the equation as the coefficient A, which represents the maximum displacement from the rest position. As for the period, it can be derived from the angular frequency using the formula T = 2π/ω, where T is the period and ω is the angular frequency.
The wavelength (λ) and speed (v) of the wave can also be determined from the properties of the wave. The wavelength is related to the wave number by the equation λ = 2π/k, and the speed can be calculated if the frequency (f) is known, using the equation v = λf.
For example, if the amplitude is provided as 30 V/m and the frequency is given as 2.0 MHz, the period (T) would then be 5.0 × 10-7 s based on the relationship between frequency and period, T = 1/f.
Ariel has a standard deck of 52 cards. She draws one card from the deck and then replaces it. She then draws a second card from the deck. What is the probability that both cards are clubs? B) Alisa has a bag of 10 red and 6 green marbles. She takes one marble out of the bag and does not replace it. She then takes a second marble out of the bag. What is the probability that both marbles are green? C) In a bag of apples there are 4 green apples and 2 red apples. Bethany takes a green apple out of the bag and does not replace it. She then takes another apple out of the bag. What is the probability that the second apple is red? D) In Mrs. Brown's Geometry class there are 15 boys and 10 girls. The students are presenting projects to th
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B) Like part a, the initial probability of finding a green marble is 6/16, or 0.375 (37.5%). When not replacing the marble, the second probability is now 5/15, or 0.33 (33.3%), due to the loss of this marble. This means that the cumulative probability is 0.375 x 0.333, or 0.1249 (12.49%).
C) The initial probability of finding a green apple is 4/6, or 0.66 (66.6%). The second probability, when not replacing the apple is now 2/5, or 0.4 (40%) - the loss of the apple affected the total number of apples, but didn’t affect the number of red apples. This means that the cumulative probability is 0.666 x 0.4, or 0.266 (26.6%).
D) Unfortunate, I don’t know what the problem is asking, so I can’t answer this for you.
Answer:
A)
Step-by-step explanation:
If each person takes up 2.5 square feet of space, how many people can fit in an elevator that is 8 feet by 6 feet? Round to the nearest person.
2.5
19
38
48
Answers
To get that you would multiply 8x6=48
So now since each person takes 2.5 square feet you would divide 48 by 2.5
48/2.5=19.2
19.2 rounded to the nearest whole number is 19
Your final answer is 19..
How is the graph of log x + 6 translated from the graph of log x
Answers
A truck whose bed is 2.5 m long, 1.5 m wide and 1.0 m high is delivering sand for a sand-sculpture competition. About how many trips must the truck make to deliver 7 m3 of sand?
Answers
What are the trigonometric ratios of a right triangle ABC that has its right angle at C? AC=12 and BC=5
Answers
Sine = opposite/hypotenuse,
Cosine = adjacent/hypotenuse and
tangent = opposite/adjacent
Since in the question we have not been given the hypotenuse, we calculate it first.
So, hypotenuse = √(12²+5²) = √(144+25) = √169 = 13.
Now we can find the trigonometric ratios;
Sin A = Cos B = 5/13
Sin B = Cos A = 12/13
Tan A = 5/12
Tan B= 12/5
Aiko stands 5 m from a tree. At that distance, the angle of elevation from the ground to the top of the tree is 80 degrees. Approximately how tall is the tree? Enter the value rounded to the nearest tenth. Do not include the units of measure.
Answers
a² + b² = c²
Now, just plug in the numbers.
a² + 5² = 80²
So, a² + 25 = 6400
Now, just perform the order of operations on both sides of the equation.
a² + 25 = 6400
-25 -25
------ -------
a² = 6375
√a² = √6375
a = 79.8 meters, if the hypotenuse, or slant height of the boy to the tree is 80m.
If kaitlyn goes to the store, the probability that she buys blueberries is 90%. the probability of her going to the store is 30%. what is the probability of her going to the store and buying blueberries?
Answers
P(going to the shop)*P(Buying blue berries)
let P(going to the shop)=P(A)=0.3
P(Buying blue berries)=P(B)=0.9
P(A and B)=P(A)∩P(B)
=0.3×0.9
=0.27
Answer: 0.27
Final answer:
The answer explains the calculation of the probability of an event occurring using provided probabilities and the intersection formula.
Explanation:
The probability of Kaitlyn going to the store and buying blueberries can be calculated using the formula for the intersection of two events.
We know that P(going to the store) = 0.3, P(buying blueberries) = 0.9, and we are looking for P(going to the store and buying blueberries).
Therefore, the probability of Kaitlyn going to the store and buying blueberries is 0.3 × 0.9 = 0.27 or 27%.
How to find two numbers that sum equal 15 and product are maximized?
Answers
f(x) = -x^2 + 15x
The maximum value is in the vertex of parabola (h; k).
For ax² + bx + c:
[tex]h=\dfrac{-b}{2a};\ k=f(h)=\dfrac{b^2+4ac}{4a}[/tex]
We have: a = -1; b = 15; c = 0
substitute:
[tex]h=\dfrac{-15}{2\cdot(-1)}=\dfrac{15}{2}=7.5[/tex]
15 - 7.5 = 7.5
Answer: 7.5 and 7.5.
True or False? A triangle with two congruent angles can never be scalene. A. True B. False
Answers
Hive a nice day! ♪
Hope this helped
it took Otis 6 hours to travel to the Grand Canyon. along the way he took 18 minutes to get gasoline and 53 minutes to eat. how much time did Otis spend driving?