By the principle of mathematical induction, the statement holds for all positive integers n.
How did we arrive at this assertion?Using mathematical induction:
Base case:
For n=1, results into:
2 = 2^2 + 1 - 2
which is true.
Inductive step:
For some positive integer k, we have:
2+4+8+...+2^k = 2^(k+1) + 1 - 2
This implies the statement for n=k+1, i.e.,
2+4+8+...+2^k+2^(k+1) = 2^(k+2) + 1 - 2
From the left-hand side of the equation, we can rewrite it as:
2+4+8+...+2^k+2^(k+1) = (2+4+8+...+2^k) + 2^(k+1)
Applying the induction hypothesis, substitute the expression for 2+4+8+...+2^k:
2+4+8+...+2^k+2^(k+1) = (2^(k+1) + 1 - 2) + 2^(k+1)
Simplify:
2+4+8+...+2^k+2^(k+1) = 2^(k+2) - 1
Using the formula for the sum of a geometric series to simplify the right-hand side of the original statement:
2^(k+2) + 1 - 2 = 2^(k+2) - 1
Thus, the statement holds for n=k+1, supposing it holds for n=k. By the principle of mathematical induction, the statement holds for all positive integers n.
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LINES AND ANGLES
Circle the correct answer to each of the following questions. Show your work, if necessary.
1 Which of the following objects is two-dimensional?
a. a pair of dice
b. a basketball
C
tree
d. a circle
Which expression is equivalent to
1/sin(2x)-cos(2x)/sin(2x)?
The trigonometric expression 1/sin(2x) - cos(2x)/sin(2x) = tanx
What is a trigonometric expression?A trigonometric expression is an equation that contains trigonometric ratios.
Given the expression 1/sin(2x) - cos(2x)/sin(2x), we need to find the expression that is equivalent to it.
So, we proceed as follows.
1/sin(2x) - cos(2x)/sin(2x) = [1 - cos(2x)]/sin(2x),
Using the trigonometric identity cos2x = cos²x - sin²x and sin2x = 2sinxcosx, we have that
[1 - cos(2x)]/sin(2x) = [1 - (cos²x - sin²x)]/2sinxcosx
= [1 - cos²x + sin²x)]/2sinxcosx
Now, 1 - cos²x = sin²x.
So, substituting this into the equation, we have that
[1 - cos²x + sin²x)]/2sinxcosx = [sin²x + sin²x)]/2sinxcosx
= [sin²x + sin²x)]/2sinxcosx
= 2sin²x/2sinxcosx
= sinx/cosx
= tanx
So, 1/sin(2x) - cos(2x)/sin(2x),= tanx
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the lowest common multiple of 36 30
Answer:
180
Step-by-step explanation:
To find the lowest common multiple (LCM) of 36 and 30, we can use the prime factorization method:First, we can find the prime factorization of each number:36 = 2^2 * 3^230 = 2 * 3 * 5Next, we can take the highest power of each prime factor that appears in either number and multiply them together.The highest power of 2 is 2^2 = 4.The highest power of 3 is 3^2 = 9.The highest power of 5 is 5^1 = 5.Multiplying these together, we get:LCM(36, 30) = 2^2 * 3^2 * 5 = 180Therefore, the lowest common multiple of 36 and 30 is 180.
Answer: 180
Step-by-step explanation:
To find the lowest common multiple (LCM) of two numbers, we need to find the smallest number that is a multiple of both of them.
One way to do this is to find the prime factorization of each number and then take the product of all the prime factors, with each factor occurring as many times as it appears in the factorization of either number.
The prime factorization of 36 is 2^2 × 3^2, and the prime factorization of 30 is 2 × 3 × 5. Therefore, the LCM of 36 and 30 can be found by taking the product of the highest power of each prime factor that appears in either factorization:
LCM(36, 30) = 2^2 × 3^2 × 5 = 180
Therefore, the lowest common multiple of 36 and 30 is 180.
The Sky Train from the terminal to the rental car and long term parking center is supposed to arrive every 8 minutes. The waiting times for the train are known to follow a uniform distribution.
What is the probability of waiting less than 2 minutes and more than 6 minutes?
The 30th percentile in the uniform distribution is 2.4 min.
The correct option is (b)
Probability :The probability formula is defined as the ratio of favorable outcomes to the ratio of total outcomes. For any event (E), :
P(E)= Number of favorable outcomes/ Number of total outcomes
[tex]\mu=\frac{a +b}{2}[/tex]
[tex]\sigma=\sqrt{\frac{(b-a)^2}{12} }[/tex]
P(x) = 1/ b - a if a < x < b
P(X < a) = 0
and
P(X > b) = 0
In this case, a = 0 and b = 8
The 30th percentile in the uniform distribution, this means that the probability is 0.30. Therefore:
0.3 = [tex]\frac{1}{8-0}.x[/tex]
0.3 (8 -0) = x
2.4 = x
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The given question is incomplete, complete question is :
The Sky Train from the terminal to the rental-car and long-term parking center is supposed to arrive every eight minutes. The waiting times for the train are known to follow a uniform distribution. Find the 30th percentile for the waiting times (in minutes).
a. 2
b. 2.4
c. 2.75
d. 3
For questions 11-12, divide. Use synthetic division, if possible.
11. (x² +6x²-x² - 5x+1)+(x-2)
12. (x³ +9x² -5x +11)+(x² + 2)
Question 11 seems to have strange typos in it. Please double-check to make sure it has been typed correctly.
I'm not sure how to answer that one.
----------
I'll focus on Question 12.
The long division scratch work is shown in the image attachment below.
The answer to problem 12 is x+9 remainder -7x-7
Please let me know if you have questions about any step of the polynomial long division process.
Find the surface area of the cylinder in terms of Pi. The diameter is 14 cm and the height is 18 cm
The surface area of the cylinder in terms of pi is:
S = pi*301 cm²
how to find the surface?We know that for a cylinder of radius R and height H the surface area is:
S = pi*( 2RH + R²)
Here the height is 18 cm and the diameter is 14cm, then the radius is.
R = 14cm/2 = 7cm
Then the surface area will be:
S = pi*(2*7cm*18cm + (7cm)*7cm)
S = pi*301 cm²
That is the surface area in terms of pi.
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Given f(x) = 7 - 4x and f(x) = 27, what is the value of x?
Answer: x = -5
Step-by-step explanation:
Answer:
x=-5
7 - 4 * (-5)
Negative times a negative equals a positive
-4*-5=20
7+20=27
Victoria wants to figure out if she can run faster than
an elephant. She reads that an elephant can run 100
yards in 20 seconds. She knows she can run 60 feet in
10 seconds. Victoria says, "Since 60/10 = 6 and 100/20 = 5,
my speed is faster than an elephant's!" Does her
statement make sense? Explain why or why not.
20
Answer and explanation:
Victoria's statement that her running speed is faster an elephant's does not make sense because it is based on the comparison of unlike units of measurement.
How the above conclusion is reached:Elephant's running speed = 100 yards in 20 seconds
= 5 yards per second (100 ÷ 20)
Units of Measurement:1 yard = 3 feet
5 yards = 15 feet (5 x 3)
The elephant's running speed = 15 feet per second.
Victoria's running speed = 60 feet in 10 seconds
= 6 feet per second (60 ÷ 10)
Thus, we conclude that the elephant's running speed is 15 feet per second, while Victoria does 6 feet per second, making her claim untrue.
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A jelly bean is selected at random from ajar and is not returned. The jar contains 8 red, 2 green, 4 blue, and 7 pink, jelly beans. Which is the probability of selecting a ed jelly bean on the first draw and a blue jelly bean on the second draw?
Answer:
There would be a 38% or 8/21 probability of picking a red jelly bean on the first draw and a 20% or 4/20 probability of picking a blue jelly bean on the second.
Step-by-step explanation:
There are 21 jelly beans in total. So on the first draw, there is an 8/21 chance you will get a red jelly bean. If you turn this fraction into a percentage, it is 38%.
Now that a jelly bean is missing, there are 20 jelly beans. There are four blue ones still, so there is a 4/20 chance of getting a blue jelly bean. Turning this ratio into a percent gets you 20%.
I would love a brainliest. I haven't had one yet and it would make my day. Hope this helped you!
The probability of selecting a red jelly bean on the first draw and a blue jelly bean on the second draw is 32/441.
What is Probability?Probability refers to potential. A random event's occurrence is the subject of this area of mathematics.
The range of the value is 0 to 1. Mathematics has incorporated probability to forecast the likelihood of various events.
The degree to which something is likely to happen is basically what probability means.
We have,
The jar contains 8 red, 2 green, 4 blue, and 7 pink, jelly beans.
total beans = 8 + 2 + 4 + 7 = 21
So, the probability of selecting a red jelly bean on the first draw and a blue jelly bean on the second draw
= 8/21 x 4 / 21
= 32 /441
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IQR of 28 22 15 16 15 13 19 18
Answer: The IQR of this data set is 5.5
Step-by-step explanation: Ok so first you line the data set up in correct order.
Here's the correct order:
13, 15, 15, 16, 18, 19, 22, 28
Then you find the median and since there's even amount of numbers you add the two numbers in the middle which is 16 and 18 and divide that by 2 which is 17. So the median is 17. Then you split the data set into halves. The first quartile is 15 and the third quartile is 20.5. Then to find the IQR, you would minus 20.5 - 15 = 5.5. Hope this helps!
Select the smaller fraction. 1/10 or 1/2
Answer:
1/10
Step-by-step explanation:
Select the smaller fraction. 1/10 or 1/2
1/10 = 1 : 10 = 0.1
1/2 = 1 : 2 = 0.5
so your answer is 1/10
PLEASE HELP ME SOLVE THIS!!!!
The meaning of the absolute value of -5.5 is given as follows:
The temperature decreases 5.5 degrees.
What is the absolute value of a number?The absolute value of a number is the number without the signal, for example:
|-2| = |2| = 2.
The number in this problem is given as follows:
-5.5.
Hence:
The negative sign means that the temperature decreased.The absolute value of 5.5 means that the temperature decreased by 5.5 degrees.Hence the second option is the correct option in the context of this problem.
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A train moving 50 miles per hour meets and passes a train moving 50 miles per hour in the opposite direction. A passenger in the first train sees the second train pass in 5 seconds. How long is the second train?
Answer:
The length of the second train is 416.67 meters.
Step-by-step explanation:
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In circle R, find arc length of arc GH
The length of the arc GH is 13.3 cm
What is length of an arc?The length of an arc is the distance that runs through the curved line of the circle making up the arc.
The length of an arc is expressed as;
l = tetha/360 × 2πr
tetha = R
R = 360-( 170+80)
R = 360-250
R = 110°
l = 110/360 × 2 × 3.14 × 6.4
l = 4787.2/369
l = 13.3 cm (1.dp)
therefore the length of the arc GH is 13.3 cm
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Cindy has a new job offer but will need a new car for the job. After planning a budget, they determine that they can afford to pay at most $215 per month for a 6-year car loan. If an annual percentage rate of 2.1% is available to finance the car loan, calculate the value of the most expensive car loan that Cindy can afford. Round to the nearest whole
number
The value of the most expensive car loan that Cindy can afford is $13,712
We are given that;
Amount= $215
Percentage= 2.1%
PMT = 215 i = 0.021 / 12 = 0.00175 (the annual interest rate divided by 12 months) n = 6 × 12 = 72 (the number of months in 6 years)
Now,
The formula for the present value of an annuity is:
PV = PMT × (1 - 1/(1 + i)^n) / i
where:
PV = present value PMT = payment amount i = interest rate per period n = number of periods
Plugging these values into the formula, we get:
PV = 215 × (1 - 1/(1 + 0.00175)^72) / 0.00175 PV = 215 × (1 - 0.888) / 0.00175 PV = 215 × 0.112 / 0.00175 PV = 13,712
Therefore, by the percentage the answer will be $13,712.
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A fair die is rolled 4 times. What is the probability of having no 1 and no 4 among the rolls? Round your answer to three decimal places.
Solve the equation.
22
= 6
2 =
The number of runs scored by the Stars for six games is shown below.
4,9, 2, 4, 1,6
If the Stars scored 14 runs in their seventh game, which of the following statements is true?
OA. The mean increases and the median remains the same.
OB. The median increases and the mean remains the same.
OC. The mean and the median both increase.
O D. The median and the mean both remain the same.
Jake and Rafael are training for a bike race. Jake is a beginner and starts by riding up a hill with a 7.9° incline. Rafael is more experienced and rides up a hill with an 18.3° incline. If Jake and Rafael ride 100 ft., how much more horizontal distance did Jake cover?
Responses
4.1 ft.
4.1 ft.
9.6 ft.
9.6 ft.
17.7 ft.
17.7 ft.
19.2 ft.
The horizontal distance more that Jake was able to cover, given the incline, would be A.4.1 ft.
How to find the horizontal distance ?We can find the horizontal distance that both Jake and Raphael covered .
Jake's distance was:
cos ( 7.9 ° ) = adjacent / 100
adjacent = 100 x 0.989
adjacent = 98.9 ft
Rafael's distance was :
cos ( 18. 3° ) = adjacent / 100
adjacent = 100 x 0.951
adjacent = 95.1 ft
The difference and the distance covered in addition by Jake would be:
= 98. 9 - 95. 1
= 3. 8 ft
The closest option is 4. 1 ft.
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Which equation describes a vertical translation of the square root parent function?
A. y = x − 4−−−−−−√
x
−
4
B. y = x−−√−6
x
-
6
C. y = x−−√
x
D. y = −x−−√
Answer:
The equation that describes a vertical translation of the square root parent function is A. y = x − 4−−√ + k where k is the vertical shift.
The square root parent function is f(x) = √x. To perform a vertical translation of this function, we add or subtract a constant value to the function. In this case, the function y = x − 4−−√ represents a vertical translation of the square root parent function by 4 units downwards.
Option B, y = x−−√−6 represents a vertical translation of the square root parent function by 6 units downwards. Option C, y = x−−√, represents the square root parent function without any vertical shift. Option D, y = −x−−√, represents a reflection of the square root parent function about the y-axis.
The equation of a line is y= -2/7x + 3/7. What is the slope of a line perpendicular to this line?
The slope of the line perpendicular to the line is m = ( 7/2 )
Given data ,
The equation of the given line is y = -2/7x + 3/7.
Now , it is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
The slope of the given line is -2/7, which means that for any line perpendicular to this line, the slope would be the negative reciprocal of -2/7.
So , m = ( 7/2 )
Hence , the slope of the perpendicular line is = 7/2
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What is 2x-5y=9 in slope intercept form?
The dimensions of square A are three times the dimensions of square B. The area of square B is 64 cm what is the area of square A
A
The bottom of this cylinder, called the
base, has an area of 22 in.² It takes
22 in. of sand to cover the base.
The cylinder has a height of 3 in.
How much sand can the cylinder hold?
B
Which can be used to find the volume of a cylinder?
v=Area of the base + height of the cylinder
V= (Area of the base)2 x height of the cylinder
v=Area of the base x height of the cylinder
V = Area of the base
Answer: 1) 66 2)
Step-by-step explanation:
1) 22x3=66
2) C
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Explain how to determine the zeros of g (x) = -(x+2) (x-3) (x-3)
The zeros of g(x)= -(x+2) (x-3) (x-3) are -2 and 3.
We are given that;
g (x) = -(x+2) (x-3) (x-3)
Now,
To determine the zeros of g(x) = -(x+2)(x-3)(x-3), we need to find the values of x that make g(x) equal to zero. This means that we need to solve the equation:
−(x+2)(x−3)(x−3)=0
To solve this equation, we can use the zero product property, which states that if the product of two or more factors is zero, then at least one of the factors must be zero. Therefore, we can set each factor equal to zero and solve for x:
x+2=0orx−3=0
Subtracting 2 from both sides of the first equation, we get:
x=−2
Adding 3 to both sides of the second equation, we get:
x=3
Therefore, by the given equation the answer will be -2 and 3.
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Need help with his page 20 points
Answer:
m/2 -6= m/4+2 can be solved as follows:
Multiply both sides of the equation by the least common multiple of the denominators, which is 4:
4(m/2 - 6) = 4(m/4 + 2)
2m - 24 = m + 8
Subtract m from both sides:
m - 24 = 8
Add 24 to both sides:
m = 32
Therefore, the value of m is C) 32.
k/12 = 25/100 can be solved as follows:
Multiply both sides of the equation by 12:
k = 12 * (25/100)
k = 3
Therefore, the value of k is A) 3.
9/5 = 3x/100 can be solved as follows:
Multiply both sides of the equation by 100:
100 * (9/5) = 3x
Simplify:
180/5 = 3x
36 = 3x
Divide both sides by 3:
x = 12
Therefore, the value of x is not one of the options provided.
Step-by-step explanation:
Answer:
Question 18:-[tex] \sf \longrightarrow \: \frac{m}{2} - 6 = \frac{m}{4} + 2 \\ [/tex]
[tex] \sf \longrightarrow \: \frac{m - 12}{2} = \frac{m + 8}{4}\\ [/tex]
[tex] \sf \longrightarrow \: 4(m - 12) =2(m + 8)\\ [/tex]
[tex] \sf \longrightarrow \: 4m \: - 48 =2m + 16\\ [/tex]
[tex] \sf \longrightarrow \: 4m \: - 2m = 16 + 48\\ [/tex]
[tex] \sf \longrightarrow \:2m = 64\\ [/tex]
[tex] \sf \longrightarrow \:m = \frac{64}{2} \: \\ [/tex]
[tex] \sf \longrightarrow \:m = 32 \: \\ [/tex]
[tex] \qquad{ \underline{\overline{ \boxed{ \sf{ \: \: C) \: \: \: 32 \: \: \: \: \: \checkmark }}}}} \: \: \bigstar[/tex]
__________________________________________
Question 19:-[tex] \sf \leadsto \: \frac{k}{12} = \frac{25}{100} \\ [/tex]
[tex] \sf \leadsto \: 100(k)= 12(25) \\ [/tex]
[tex] \sf \leadsto \: 100 \times k= 12 \times 25 \\ [/tex]
[tex] \sf \leadsto \: 100 k= 300 \\ [/tex]
[tex] \sf \leadsto \: k= \frac{300}{100} \\ [/tex]
[tex] \sf \leadsto \: k= 3 \\ [/tex]
[tex] \qquad{ \underline{\overline{ \boxed{ \sf{ \: \: a) \: \: \: 3 \: \: \: \: \: \checkmark }}}}} \: \: \bigstar[/tex]
__________________________________________
Question 20:-[tex] \sf \longrightarrow \: \frac{9}{5} = \frac{3x}{100} \\ [/tex]
[tex] \sf \longrightarrow \: 100(9)= 5(3x) \\ [/tex]
[tex] \sf \longrightarrow \: 100 \times 9= 5 \times 3x \\ [/tex]
[tex] \sf \longrightarrow \: 900= 15x \\ [/tex]
[tex] \sf \longrightarrow \: x= \frac{900}{15} \\ [/tex]
[tex] \sf \longrightarrow \: k= 60 \\ [/tex]
[tex]\qquad{\underline{\overline {\boxed{ \sf{ \: \: a) \: \: \: 60 \: \: \: \: \: \checkmark }}}}} \: \: \bigstar[/tex]
__________________________________________
A bakery sold 107 cupcakes in one day. The head baker predicted he would sell 87 cupcakes that day. What was the percent error of the baker's prediction?
Answer:
First, we need to calculate the absolute error, which is the difference between the predicted value and the actual value:
Absolute error = |predicted value - actual value| = |87 - 107| = 20
Then, we can calculate the percent error using the formula:
Percent error = (absolute error / actual value) x 100%
Plugging in the values, we get:
Percent error = (20 / 107) x 100% ≈ 18.69%
Therefore, the percent error of the baker's prediction is approximately 18.69%.
The length of one of the legs of a right triangle is 17. The lengths of the other two sides are consecutive integers. Use the Pythagorean theorem to solve for the smaller of the two missing sides (the second leg).
The value of the smaller of the two missing sides (the second leg) is,
⇒ 144 units
We have to given that;
The length of one of the legs of a right triangle is 17.
And, The lengths of the other two sides are consecutive integers.
Hence, Let the smallest of the other legs be x,
And, the hypotenuse be x + 1.
So, By using the Pythagorean theorem as;
⇒ 17² + x² = (x + 1)²
⇒ 289 + x² = x² + 1 + 2x
⇒ 2x = 288
⇒ x = 144
Thus, The value of the smaller of the two missing sides (the second leg) is,
⇒ 144 units
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It's a beautiful fall day! Mari and her friends decide to have a picnic in the park. Each friend is bringing something to eat or drink. Mari is bringing watermelon. Before she leaves her house, Mari cuts a watermelon into 16 equal pieces. At the picnic, she eats 3 pieces of watermelon.What fraction of the watermelon does Mari eat?
The 5-lb collar slides on the smooth rod, so that when it is at A it has a speed of 10 ft/s. A) if the spring to which it is at- tached has an unstretched length of 3 ft and a stiffness of k-= 10 lb/ etermine the normal force on the collar at this instant. B)Determine the acceleration of the collar at this instant.
The acceleration of the collar at point A is 5 ft/s^2.
A) To determine the normal force on the collar at point A, we need to consider the forces acting on the collar. The only force acting on the collar in the vertical direction is the weight of the collar (5 lb), which is balanced by the normal force exerted by the rod. Therefore, we can write:
N - 5 = 0
where N is the normal force. Solving for N, we get:
N = 5 lb
B) To determine the acceleration of the collar at point A, we need to use Newton's second law, which states that the net force acting on an object is equal to its mass times its acceleration. The net force on the collar is given by the force exerted by the spring, which is equal to the spring constant times the displacement of the collar from its unstretched length. At point A, the displacement of the collar is:
x = L - y = 3 - 0 = 3 ft
where L is the length of the rod and y is the position of the collar on the rod. Therefore, the force exerted by the spring is:
F = kx = 10 lb/ft × 3 ft = 30 lb
The weight of the collar is:
W = mg = 5 lb
where g is the acceleration due to gravity. The net force on the collar is therefore:
Fnet = F - W = 30 - 5 = 25 lb
Using Newton's second law, we can write:
Fnet = ma
where a is the acceleration of the collar. Solving for a, we get:
a = Fnet / m = 25 lb / 5 lb = 5 ft/s^2
Therefore, the acceleration of the collar at point A is 5 ft/s^2.
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