a) The derivative of f(x) = x^4 + 4x is f'(x) = 4x^3 + 4.
b) The antiderivative of f'(x) = 4x^3 + 4 is f(x) = (1/4)x^4 + 4x + C
What is derivative of function ?
The derivative of a function of a real variable measures the function's sensitivity to change in relation to a change in its input. Calculus relies heavily on derivatives.
a) To find the derivative of f(x) = x^4 + 4x, we can use the power rule and the linear rule of differentiation. The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1). The linear rule states that if f(x) = cx, then f'(x) = c. Applying these rules, we have:
f'(x) = (x^4)' + (4x)'
= 4x^3 + 4
So, the derivative of f(x) = x^4 + 4x is f'(x) = 4x^3 + 4.
b) The question is asking for f(x), not f'(x). To find f(x), we need to integrate the derivative f'(x) with respect to x. We can use the power rule and the linear rule of integration to find an antiderivative for f'(x). The power rule states that if f'(x) = x^n, then f(x) = (1/(n+1))x^(n+1) + C, where C is an arbitrary constant of integration. The linear rule states that if f'(x) = c, then f(x) = cx + C. Applying these rules, we have:
f(x) = ∫(4x^3 + 4)dx
= (1/4)x^4 + 4x + C
So, The antiderivative of f'(x) = 4x^3 + 4 is f(x) = (1/4)x^4 + 4x + C, where C is an arbitrary constant of integration.
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How does counting cubes compare to multiplying dimensions when finding the volume of a rectangular prism?
Select from the drop-down menus to correctly complete the statements.
Amelie correctly determined the volume of this prism by counting unit cubes. She counted
Choose...
unit cubes.
Eloise correctly determined the volume of this prism by multiplying the three dimensions. She got a product of
Choose...
cubic units.
Amelie and Eloise correctly concluded that counting unit cubes gives
Choose...
result as multiplying the dimensions.
Eloise correctly determined the volume of this prism by multiplying the three dimensions, She got a product of (3×8×4) unit³.
What is a cuboid?A cuboid is a three-dimensional closed figure which has volume along with surface area.
The volume of a cuboid is the product of its length, width, and height.
The total surface area of a cuboid is 2(lw + wh + wl) and the lateral surface area is 2(l + w)×h.
The given figure is a cuboid made up of unit cubes.
We know the volume of a cuboid is (length×width×height).
By observing the figure we have 3 unit cubes of width, 8 unit cubes of length, and 4 unit cubes of height.
Therefore, The volume of the cuboid is,
= (3×8×4) unit³.
= 96 unit³.
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Question content area top
Part 1
Error Analysis You and a friend are doing math homework together. You have to solve the equation 5x + 4x - 68 = 34 - 8x . Your friend arrives at the incorrect answer x = -2 . Find the correct value of x. Then find the error your friend possibly made.
What possible error could your friend have made to get x = -2 ?
A.
Your friend should have added 68 to each side of the equation, but instead subtracted 68 from the right side.
B.
Your friend should have added 5x and 4x to get a sum of 9x, but instead got a sum of 8x.
C.
Your friend should have added 8x to each side of the equation, but instead subtracted 8x from the left side.
The value of x is 6
The possible error could your friend have made to get x = -2 is Your friend should have added 5x and 4x to get a sum of 9x, but instead got a sum of 8x. Option C
What are algebraic expressions?Algebraic expressions are described as expressions that are composed of variable, terms, coefficients, factors and constants.
These expressions are also made up of mathematical or arithmetic operations. These operations include;
SubtractionDivisionMultiplicationFloor divisionAdditionBracketParenthesesFrom the information given, we have the equation;
5x + 4x - 68 = 34 - 8x
collect like terms
5x + 4x + 8x = 34 + 68
Add or subtract the like terms
17x = 102
Divide both sides by the coefficient of the variable, we get;
17x/17 = 102/17
x = 6
Hence, the value is 6
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Laine correctly factor expreion 16. 5 a - 77. 2 b 36 and give two poible anwer Laine could have written
The two possible answer is 3(5.5a - 2.4b + 12) that Laine could have written for expression 16.5a - 77.2b + 36 .
Laine could just just used another common number to write it down as,
LCM(165, 36, 72) = 3960
Prime factorization of the numbers:
165 = 3 × 5 × 11
36 = 2 × 2 × 3 × 3
72 = 2 × 2 × 2 × 3 × 3
LCM(165, 36, 72)
= 2 × 2 × 2 × 3 × 3 × 5 × 11
= 3960
GCD(165, 36, 72) = 3
So we get three as a common multiple of them so after removing 3 common from them we can write 16.5a - 7.2b + 36 as
3(5.5a - 2.4b + 12)
Hence, The two possible answer is 3(5.5a - 2.4b + 12) that Laine could have written for expression 16.5a - 77.2b + 36.
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Find k such that x(t)=3^t is a solution of the differential equation dx/dt = kx
The value of k such the the solution of differential equation of dx/dt = kx is k = ln(3).
What is differential rule of exponential function?By multiplying the original exponential function by the derivative of its power, one can differentiate an exponential.
The rule for differentiating an exponential function is f(x)=a^u = a^t ln(t).
The given function is:
x(t) = 3^t
Using the differential rule for the exponential function we have:
d/ dt (3^t) = 3^t ln(3)
Substituting the value of x(t) in the equation with 3^t ln(3) we have:
3^t ln(3) = k 3^t
Since 3^t is a non-zero expression the value of k corresponding to the given value is:
ln(3) = k
Hence, the value of k such the the solution of differential equation of dx/dt = kx is k = ln(3).
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Assume you have a future liability of $12000 per year for 4 years beginning 7 years from today. You will fund this liability over the next 5 years, with the first deposit to occur 1 year from today. If you earn an interest rate of 8% per year, compounded annually, on deposits, how much will you have to deposit at the end of each of the next 5 years to fund this liability?
$6273.04 have to deposit at the end of each of the next 5 years to fund this liability
PV of Annuity = P*[1-{[tex]{1+i}^{n}[/tex]}]/i
Where, P = Annuity = 12000, i = Interest Rate = 0.08, n = Number of Periods = 4
PV at 6 years from today = 12000*[1-{(1+0.08)^-4}]/0.08 = 12000*0.26497/0.08 = $39745.52
PV at 5 years from today = PV at 6 years from today/[1+Interest Rate] = 39745.52/[1+0.08] = $36801.41
FV of Annuity = P*[{[tex]1+i^{n}[/tex]}-1]/i
Where, FV = 36801.41, i = Interest Rate = 0.08, n = Number of Periods = 5
Therefore,
36801.41 = P*[{(1+0.08)^5}-1]/0.08
2944.1127 = P*0.469328
Therefore, Amount to be deposited =
P = 2944.1127/0.469328 = $6273.04
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at 9:30 a.m., sean started filling swimming pool. At 11:30a.m.,, he had filled 1,800 gallons. at what time will the pool be full?
At 2:30 AM,
the pool will be full.
What is division?The division in mathematics is one kind of operation. In this process, we split the expressions or numbers into the same number of parts.
At 9:30 AM,
Sean started filling a 4,500-gallonswimming pool.
At 11:30 AM,
he had filled 1,800 gallons.
The unit rate = 1800/2 = 900 gallons per hour.
The pool completely filled in 5 hours.
Therefore, the pool be full at 2:30 AM.
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The complete question:
At 9:30 AM,
Sean started filling a4,500-gallonswimming =~ pool. At 11:30 a.m., he had filled 1,800 : gallons. At what time will the pool be full? :
Deepa invests money in an account paying a simple interest of 1% per year. If no money will be added or removed from the investment, what should she multiply her current balance by to find her total balance in a year in one step?
For Deepa to find his yearly balance in a single step, he must multiply his money by 1.01
Comparing basic and compound interest ?Simple interest is often a predetermined percentage of the principle amount borrowed or lent paid or received over a specific time period. Borrowers are required to pay interest on interest in addition to principal since compound interest accrues and is added to the accrued interest from prior periods.
The following calculation must be done in order to determine what Deepa should multiply his current balance by in order to find his total balance in a year in one step if no money will be added to or removed from the investment. Deepa invests money in an account that pays a simple interest rate of 1% per year.
Current balance = X
Interest = 1% = 1/100 = 0.01
Interest + principal = 1.01
1.01X = Total balance in a year
Deepa must therefore multiply her money by 1.01 in order to determine it in a single step.
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LN = LN because of ____
Answer:
Reflexive Property
Answer:
reflexive property
Step-by-step explanation:
We can use the reflexive property to state that LN = LN.
It states that a value is equal to itself.
What is 15 meters to feet?
Answer: It is approximately 49.2126 or 49, for short.
Step-by-step explanation:
For a more approximate result, multiply the length value by 3.281
49.21 feet or 15 meters are the same.
What is the length of a mile in meters?This is due to the fact that 1,609.344 meters, or one mile, are equivalent.The Hollywood Sign is comparable in height to it. Aproximately 15 meters tall is The Hollywood Sign. (Also known as Hollywoodland Sign) The Hollywood Sign has 15 m-tall white letters that are each individual.The average person can anticipate to walk a mile in 15 to 22 minutes, according to data from a 2019 study spanning five decades. The Centers for Disease Control and Prevention estimate that people typically walk between 2.5 and 4 mph.49.21 feet or 15 meters are the same.The Hollywood Sign is comparable in height to it. Aproximately 15 meters tall is The Hollywood Sign.To learn more about meters refer to:
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24) Find the quotient and write the answer in standard form.
(3.6×10 to the power of 10) divided by 5
Answer: Quotient is 3.6*10^10/5 and the standard form is 7.2*10^9
Step-by-step explanation:
Answer:
Step-by-step explanation:
3.6×10∧10÷5=0.72×10∧10
standard form=7.2×10∧9
Bonsoir es ce que quelqu'un pourrait m'aider pour cette exercice plus précisément la question 2 et 3.
Merci d'avance
Answer:
2) 1885 km
3) 795 h 46 m 29 s
Step-by-step explanation:
2)
C = 2πr
C = 2 × 3.14159 × 300 km
C = 1885 km
3)
(1000000 km)/ (1885 km/tour) = 530.16477 tours
1 tour = 1 h 30 m = 1.5 h
530.16477 tours = 530.16477 × 1.5 h = 795.7747155 h = 795 h 46 m 29 s
The function h(x) = -x +34 models the height of the roof of a house, where x is the horizontal distance from the center of the house. If a raindrop
falls from the end of the roof, how far from the center of the base does it land? Explain your solution.
The distance from the base's center at that time is 18 feet.
What is function?
A relation between a collection of inputs and outputs is known as a function. A function is, to put it simply, a relationship between inputs in which each input is connected to precisely one output. Each function has a range, codomain, and domain. The usual way to refer to a function is as f(x), where x is the input.
A function is typically represented as y = f. (x).
h(x) = 16 ft x is the distance from the base when
h(x) = 16 h(x) = -IxI + 34
when a raindrop falls from the end of the roof.
16 = -IxI + 34
IxI = 34 – 16 = 18
It might land on the roof's opposite end.
The distance from the base's center at that time is 18 feet.
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A person is watching a boat from the top of a lighthouse. The angle of depression from the person to the boat is 28°. The lighthouse is 200 feet tall. How far away is the boat? Round to the nearest foot
The boat is approximately 515 feet away from the lighthouse.
To find the distance from the boat to the lighthouse, we can use trigonometry. The angle of depression from the person to the boat is 28°, and the height of the lighthouse is 200 feet. Let's call the distance from the boat to the lighthouse x. Using the tangent function, we have
tan 28° = 200 / x.Solving for x, we have x = 200 / tan 28°. Approximating this value using a calculator, we have x ≈ 514.7 feet. Rounding to the nearest foot, the boat is approximately 515 feet away from the lighthouse.
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two sides of a triangle measure 18 and 13 meters which measure could represent the perimeter of the triangle
Therefore, when the triangle's perimeter is 38 meters and its two sides are 18 meters and 13 meters, respectively, the opposite side of the perimeter is 7 meters.
what is perimeter ?An isolated path that surrounding, delimits, or embraces a two-dimensional shape or a single-dimension length sometimes referred to as the boundary. The perimeter of either an ellipse or circle refers itself to outermost area. The perimeter measurement has many legitimate applications. This perimeter of a form is its edge's circumference. Discover how to determine the perimeter by adding the lengths of such sides of various forms. Your can always find the perimeter of a polygon by multiplying its multiplicative inverse. The perimeter of an object is the space it around. At your house, a contained garden is one illustration. The perimeter of anything is the area encircling it. For a 50 feet x 50 feet yard, a 200 foot fence is required.
given
The distance surrounding a closed-plane figure is its perimeter, or P.
P is equal to a plus b plus c, where a, b, and c are the lengths of the three sides, and let c be the third side's undetermined length.
P = 38 m, a = 13 m, and b = 18 m are the values provided. Now, to determine the third side's length c,
P = a + b + c
38m = 13m, 18m, and c
38 m = 31 m + c
38 m minus 31 m Equals 31 m minus 31 m plus c
7 m = 0 + c
7 m = c
Therefore, when the triangle's perimeter is 38 meters and its two sides are 18 meters and 13 meters, respectively, the opposite side of the perimeter is 7 meters.
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for the fractured expression use the distributive property to write an equivalent expression
3(2x + 11y)
An equivalent expression of 3(2x + 11y) is 6x + 33y.
What are the equivalent expressions?Expressions that are equivalent to the same thing, even when they have distinct appearances. When we enter the same value(s) for the variable, two algebraic expressions that are equivalent have the same value (s).
Given:
An expression,
3(2x + 11y).
The distributive property:
a(b + c) = ab + ac.
So,
3(2x + 11y) = 6x + 33y
Therefore, 6x + 33y is an equivalent expression.
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What are the coordinates of the point?
Responses
(6, −3)
(−3, 6)
(3, 6)
(−3, −6)
Answer: (-3,6)
Step-by-step explanation:
The point is 3 spaces to the left in the second quadrant on the x-axis which is negative
Then it is 6 spaces up on the y-axis which is positive
In triangle QRS the measure of angle RSQ is 117°, and the measure of angle QRS is 23°.
What is the measure of angle SQR in degrees?
Answer:
the angle of SQR = 40°
Step-by-step explanation:
in any triangle, the sum of all angles will be equal to 180°
by adding RSQ and QRS you get 140°
180°-140°=40°
The measurement of angle ∠SQR is 40 degrees
We are given that triangle QRS the measure of angle RSQ is 117°, and the measure of angle QRS is 23°.
Since we know that in any triangle, the sum of all angles will be equal to 180°
To find the total angle of any polygon, use the formula = 180(n-2)
The "n" is number of sides a polygon have. Triangle has 3 sides.
Hence,substitute as n=3 in the formula above.
Total angle in a triangle =180(3-2)= 180(1)
= 180 degrees
∠QRS+∠RSQ+∠SQR=180°
180°-140°=40°
The angle SQR= 40°
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What is the Formula of Sum of Cubes of n Natural Numbers?
The formula for the sum of cubes of the first n natural numbers is:
(1^3 + 2^3 + 3^3 + ... + n^3) = (1 + 2 + 3 + ... + n)^2
What is the formula of sum of cubes?The sum of cubes of the first n natural numbers is the sum of cubes of the numbers 1, 2, 3, ..., n.To find the formula, we can start by using the formula for the sum of the first n natural numbers: (1 + 2 + 3 + ... + n) = n(n+1)/2.We can then cube this expression: (n(n+1)/2)^3 = n^3(n+1)^3/8.Expanding the cube on the right side, we get: n^3(n+1)^3/8 = n^3(n^3 + 3n^2 + 3n + 1)/8.Combining like terms and simplifying, we get the final result: (1^3 + 2^3 + 3^3 + ... + n^3) = (1 + 2 + 3 + ... + n)^2.This formula can be used to efficiently calculate the sum of cubes of any set of natural numbers, without having to sum the cubes individually.To learn more about natural numbers refer:
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what is the slope that goes through the points (-1,4) and (14,-2)
The slope is:
-2/5Work/explanation:
To find the slope, I use the slope formula
[tex]\sf{m=\dfrac{y_2-y_1}{x_2-x_1}}[/tex]
where m = slope.
Plug in the data
[tex]\sf{m=\dfrac{-2-4}{14-(-1)}[/tex]
Simplify
[tex]\sf{m=\dfrac{-6}{14+1}}[/tex]
Simplify the denominator
[tex]\sf{m=-\dfrac{6}{15}}[/tex]
Finally, reduce the fraction to its lowest terms.
[tex]\sf{m=-\dfrac{2}{5}}[/tex]
Hence, the slope is -2/5.Natalie has $300 in a savings account that earns 15% interest per year. How much interest will she earn in 4 years?
Answer:
Below
Step-by-step explanation:
Interest = p r t p = principle = 300 r = decimal interest = .14 t = years = 4
Interest = 300 * .15 * 4 = 180 dollars
A pump fills a pool at a constant rate. At the end of 18 minutes it has filled 2 gallons of water. Which table represents the relationship between the number of minutes and the number of gallons of water in the pool?
The table can be written as,
Minutes Gallons
9 1
18 2
27 3
36 4
What is an equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign.
Example:
2x + 3 = 7 is an equation.
We have,
At the end of 18 minutes, it has filled 2.
This can be written as
y = kx
Where x is minutes and y is the number of gallons.
So,
2 = 18k
k = 2/18 = 1/9
Now,
The equation is y = (1/9)x
The table can be written as,
Minutes Gallons
9 1
18 2
27 3
36 4
Thus,
The table that represents the relationship between the number of minutes and the number of gallons of water in the pool is given above.
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How much are 1.5 pounds in kilograms?
1 pound is approximately equal to 0.45359237 kilograms. Therefore, 1.5 pounds is approximately equal to 0.45359237 x 1.5 = 0.68 kg
How to convert pounds into kilograms?It's a common conversion to make, as the pound is a unit of weight commonly used in the United States, while the kilogram is the standard unit of mass in the International System of Units (SI).Knowing the conversion from pounds to kilograms can be useful in many situations, such as when traveling to countries that use the metric system, or when converting recipes from American to metric measurements.Additionally, converting pounds to kilograms can be useful in a variety of industries, such as health and fitness, where people may need to track their weight in either unit.Keep in mind that the conversion between pounds and kilograms is not exact, as the conversion factor is an approximation.For more precise conversions, it's best to use a conversion tool or calculator to ensure accurate results.To learn more about pound to kilogram refer:
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It costs a small production company a fixed cost of $2,600 for props and costumes plus
$140 for each performance to put on plays at area schools. For each performance, the
company earns $400. Let x represent the number of performances and let y represent
the amount of dollars of expenses or income. Then the two equations graphed below
represent the expenses and income earned by the production company. How many
performances must the company put on in order to break even?
The number of performances which the company must put on in order to break even is equal to 10 performances.
What is a point of intersection?In Mathematics, a point of intersection can be defined as the location on a graph where two (2) lines intersect or cross each other, which is primarily represented as an ordered pair containing the point that corresponds to the x-coordinate (x-axis) and y-coordinate (y-axis) on a cartesian coordinate.
By critically observing the graph of the lines representing the given system of equations, we can reasonably and logically deduce that the correct solution set lies in Quadrant I and it is denoted by the point of intersection of both the x-coordinate (x-axis) and y-axis, which is given by the ordered pair (10, 4,000).
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suppose limx→0f(x)=1 and limx→0g(x)=−7. name the rule or limit law that is used to accomplish each step of the following calculation.
Step 1: limx→0[f(x)+g(x)] .Rule or Limit Law used: Sum Rule for Limits.
Step 2: limx→0f(x)+limx→0g(x) .Rule or Limit Law used: Limit of a Sum Rule.
Step 1: limx→0[f(x)+g(x)]
This first step is using the sum Rule for Limits. This rule states that the limit of the sum of two functions is equal to the sum of the limits of the individual functions. So in this case, the limit of the sum of f(x) and g(x) is equal to the sum of the limits of f(x) and g(x) individually.
Step 2: limx→0f(x)+limx→0g(x)
This second step uses the Limit of a Sum Rule. This rule states that the limit of the sum of two functions is equal to the sum of the individual limits of the functions. So in this case, the limit of the sum of f(x) and g(x) is equal to the sum of the individual limits of f(x) and g(x). This is the same result as in the first step, but using a different rule.
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Original Price Percent of Discount Sale Price
$69.80 60% $
Answer:$45.60
Step-by-step explanation:
p=%w
69.8(.60)w
/.6 /.6
116. =w
-69.8
46.5
Answer:
349/5/5
Step-by-step explanation:
(69,8) % of 60x$
simplify the expression
109.What is the inverse natural function of the cosecant?a) Secantb) Sinec) Cosined) Tangent
The inverse of the cosecant function is the secant function so option A is correct.
In mathematics, the inverse of a function "f" is another function "g" such that when "f" is applied to an input "x", the result is "x". And when "g" is applied to the result, the original input "x" is recovered. This relationship can be expressed as: f(g(x)) = x and g(f(x)) = x.
The cosecant function is defined as the reciprocal of the sine function, so csc(x) = 1/sin(x). The inverse of the cosecant function is the secant function, which is defined as the reciprocal of the cosine function, so sec(x) = 1/cos(x).
In other words, when the cosecant function is applied to an angle "x", the result is 1/sin(x), and when the secant function is applied to the result, the original angle "x" is recovered. This relationship can be expressed as: sec(csc(x)) = x and csc(sec(x)) = x.
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A farmer sells 6.9 kilograms of apples and pears at the farmer's market.
1/4 of this weight is apples, and the rest is pears. How many kilograms of pears did she sell at the farmer's market?
Answer:
1.725
Step-by-step explanation:
1/4 or .25 multiplied by the total of 6.9
A linear function is shown on the graph
What is the Romain of the function?
Check the picture below.
find the area of the trapezoid
The area of the trapezoid is 24 square units.
What is the area of the trapezoid?
A trapezoid is a quadrilateral in American and Canadian English that has at least one pair of parallel sides. In British and other dialects of English, it is referred to as a trapezium. By definition, a trapezoid in Euclidean geometry is a convex quadrilateral. Its parallel sides serve as its bases.
The area of a trapezoid can be calculated by using the formula:
Area = (a + b)h / 2,
where a and b are the lengths of the parallel sides and h is the height.
By the given figure, we can find the area of the trapezoid
Area = (a+b)h/2
Area = (4.3 + 7.7)*4/2
Area = 12*4/2
Area = 24 Square units.
Hence, the area of the trapezoid is 24 square units.
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A 16 inch pendulum swings through an angle of 40. How far does the tip of the pendulum travel in a singles swing
The tip of the pendulum travels 22.33 inches in a singles swing.
A pendulum is a weight suspended from a pivot so that it can swing back and forth under the influence of gravity. Pendulums are commonly used as timekeepers in clocks, but they also have other applications, such as in seismology, where they can be used to detect earthquakes.
Each swing does an angle of 40° in each direction, so we have a total angle of 80°.
Then, we want to get the length of an arc defined by an angle of 80° on a circle with a radius of 16 in, this is:
L = (80°/360°) * 3.14 * (2 * 16 in)
= 22.33 in.
The distance that the tip of the pendulum travels on each swing is 22.33 inches.
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The tip of the pendulum travels approximately 28.3 inches in a single swing through an angle of 40 degrees.
To calculate how far the tip of the pendulum travels in a single swing through an angle of 40 degrees, we will use the formula:
Distance = 2 * PI * Length * (angle / 360)
Where PI is approximately 3.14 and Length refers to the length of the pendulum.
In this case, the length of the pendulum is 16 inches. Therefore, we can substitute these values into the formula:
Distance = 2 * 3.14 * 16 * (40 / 360)
Distance = 2 * 3.14 * 16 * 0.111
Distance = 28.3 inches
Therefore, the tip of the pendulum travels approximately 28.3 inches in a single swing through an angle of 40 degrees.
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