For given functions f(x) and g(x), the value of (f °g)(12) is 24.
What exactly is a function?
In mathematics, a function is a rule that assigns to each input value (also called the argument) exactly one output value. The input values are typically drawn from a specific set, called the domain of the function, and the output values are drawn from another set, called the range of the function.
Now,
To find (f °g)(12), we need to first evaluate g(12) and then use the result to evaluate f(g(12)).
g(x) = √(x+4), so g(12) = √(12+4) = √16 = 4.
f(x) = 4x+8, so f(g(12)) = f(4) = 4(4) + 8 = 16 + 8 = 24.
Therefore,
The value of (f ° g)(12) = f(g(12)) = 24.
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Whitney deposited money into an account in which interest is compounded quarterly at a rate of
2. 9%. She made no other deposits or withdrawals and the total amount in her account after 12 years
was $10,891. 31.
How much did she deposit?
O
$7700
O
$8650
O
$9700
o $10,580
Next
[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\dotfill & \$ 10891.31\\ P=\textit{original amount deposited}\\ r=rate\to 2.9\%\to \frac{2.9}{100}\dotfill &0.029\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{quarterly, thus four} \end{array}\dotfill &4\\ t=years\dotfill &12 \end{cases}[/tex]
[tex]10891.31 = P\left(1+\frac{0.029}{4}\right)^{4\cdot 12} \implies 10891.31=P(1.00725)^{48} \\\\\\ \cfrac{10891.31}{1.00725^{48}}=P\implies 7700\approx P[/tex]
I NEED HELP ON 6-7 ASAP!!
Step-by-step explanation:
disculpa, te recomiendo acercar más la foto para porder
te ayudar
The reading on the thermometer is -20 degrees Fahrenheit and the temperature rises with 20 degrees Fahrenheit. What will be the new reading be?
As per the given temperature, the new reading is 0°F
The formula for converting Fahrenheit to Celsius is (°F - 32) x 5/9 = °C. We can use this formula to determine the new temperature in Celsius and then convert it back to Fahrenheit.
First we have to convert the initial temperature to Celsius
(-20°F - 32) x 5/9 = -28.89°C
Now we have to add 20 degrees Fahrenheit to the initial temperature
-20°F + 20°F = 0°F
And then we have to convert the new temperature to Celsius
(0°F - 32) x 5/9 = -17.78°C
When we convert the new temperature from Celsius back to Fahrenheit
-17.78°C x 9/5 + 32 = 0°F
Therefore, the new reading on the thermometer will be 0 degrees Fahrenheit, which indicates that the temperature has risen by 20 units on the Fahrenheit scale.
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3/5 kilogram of soil fills 1/3 of a container can 1 kilogram of soil fit
in the container
1 kilogram of soil will fit in the container, since 1 kilogram is less than 27/5 kilograms.
What is the fraction of 1 kg soil in the container?
We can start by using ratios to solve this problem.
From the information given, we know that:
3/5 kilogram of soil fills 1/3 of the containerWe want to know how much soil will fit in the entire container (which is equivalent to 3 times the amount that fills 1/3 of the container)
To find out how much soil fills the entire container, we can set up a ratio:
(3/5 kg of soil) : (1/3 of container) = (x kg of soil) : (1 container)
We can cross-multiply to solve for x:
(3/5 kg of soil) * (1 container) = (1/3 of container) * (x kg of soil)
Multiplying both sides by 3/1 gives:
(3/5) * 3 * (1 container) = (1/3) * x
Simplifying gives:
9/5 = 1/3 * x
Multiplying both sides by 3 gives:
27/5 = x
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The complete question is below:
3/5 kilogram of soil fills 1/3 of a container can, 1 kilogram of soil fit ? in the container
x= 2 cos³t y = 2 sin ³ t
find the cartesian equation
Answer:
[tex]y^2 = x^3 / 2^{3/2}[/tex] or [tex]y = x^{3/2} / 2^{3/4}[/tex]
Step-by-step explanation:
We can use the identity [tex]cos^3(t) = cos(t)*cos^2(t) and sin^3(t) = sin(t)*sin^2(t)[/tex] to rewrite the given parametric equations in terms of x and y:
[tex]x = 2cos(t)*cos^2(t) = 2x^2 - 2y^2[/tex]
[tex]y = 2sin(t)*sin^2(t) = 2xy^2[/tex]
Solving for y in terms of x, we get:
[tex]y = x^{3/2} / 2x^{3/4}[/tex]
Therefore, the cartesian equation of the curve is:
[tex]y = x^{3/2} / 2^{3/4}[/tex]
or, equivalently:
[tex]y^2 = x^3 / 2^{3/2}[/tex]
The couple asked the architect to add a square garden shed to the plans that she
drew. She added a square garden shed with coordinates (-2,8), (1,8), (1, 11), and
(-2, 11) to the plans. Discuss the following with a partner:
• How could you find the perimeter and area of the shed without graphing
the coordinates?
• How is this problem different from finding the perimeter and area of the
base of the house?
. Can you use the same strategy to solve both problems? Why or why not?
Answer:
To find the perimeter and area of the square garden shed without graphing the coordinates, you can use the distance formula to calculate the length of one side of the square. The distance formula is d = √((x2 - x1)^2 + (y2 - y1)^2), where (x1, y1) and (x2, y2) are the coordinates of two points. Using this formula with the coordinates of two adjacent corners of the square, such as (-2,8) and (1,8), we find that the length of one side of the square is 3 units. Since all sides of a square are equal in length, the perimeter is 4 times the length of one side, or 12 units. The area of a square is equal to the length of one side squared, so the area of the square garden shed is 9 square units.
This problem is different from finding the perimeter and area of the base of a house because it involves a square shape rather than a more complex shape. However, you can use similar strategies to solve both problems by calculating distances between points and using geometric formulas to find perimeter and area. The specific formulas used may vary depending on the shape of the base of the house.
Please can someone help!
Answer: 20.1 m
Step-by-step explanation:
using the formula circumference = 2pi r:
[tex]126=2\pi r[/tex]
divide 2 pi on both sides
[tex]r=\frac{126}{2\pi }[/tex]
r = 20.05
r = 20.1 m
A test has 12 problems, and each problem is worth 5 marks. Full marks are given for a correct answer, 2 marks are given if there is no answer, and no marks are given for an incorrect answer. Some scores between 1 and 60, are impossible to get on this test. What is the sum of these impossible scores?
To make things easier, consider the residue when each integer between equation 1 and 60 is divided by 5. The remainders might be one, two, three, or four.
What is equation?A math equation is a process that relates two statements by using the equals sign (=) to indicate equivalence. In algebra, an equation is a mathematical statement that proves the equality of two mathematical expressions. In the equation 3x + 5 = 14, for example, the equal sign separates the numbers 3x + 5 and 14. A mathematical formula may be used to understand the link between the two phrases written on opposite sides of a letter. Frequently, the logo and the software are the same. For example, 2x - 4 = 2.
0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 = 0 (if all answers are wrong or not attempted) (if all answers are incorrect or not attempted)
5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 60 (assuming all answers are accurate) (if all answers are correct)
10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 = 120 (assuming all answers are accurate and attempted twice) (if all answers are correct and attempted twice)
. The impossible scores may be obtained by locating all the numbers between 1 and 60 that cannot be stated as a sum of 0, 5, or 10.
To make things easier, consider the residue when each integer between 1 and 60 is divided by 5. The remainders might be one, two, three, or four.
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I need helppp do these all plsss show workk too
1. A team of 7 construction workers worked together to build 3
sheds in 10 days. How much of a shed did each of them build?
2. The construction of an 80-meter bridge was finished in 12
weeks. How much of the bridge was built on average each
week?
3. A 4-story building is 29 feet tall. Between which two whole
numbers do the height of each story lie?
1. Josh used 3 packages of spaghetti to make dinner for 19
guests. How much did each guest have?
2. According to a recipe, 190 grams of sugar is needed to make 9
cupcakes. Between which two whole numbers does the weight
of sugar used for each cupcake lie?
3. A 2-liter pitcher of juice was poured into 8 cups. How much
juice was in each cup?
There was 5
8
of a pie left in the fridge. Daniel ate 1/4 of the leftover pie. How much of a pie did he have?
2. Olivia took out 8 glasses and poured juice from the pitcher. The
capacity of each glass is 3/10
liter. If there was enough juice for 6 glasses, how much juice was there?
3. Pam baked some cupcakes for her friends. She baked 24
cupcakes. Each cupcake is 2/15
pound. If she packed 6
cupcakes in each box, what is the weight of each box?
Answer:
number 2 is 4
Step-by-step explanation:
The average high temperatures in degrees for a city are listed. 58, 61, 71, 77, 91, 100, 105, 102, 95, 82, 66, 57 if a value of 110.5° is added to the data, how does the mean change and by how much? a. the mean increases by 2.3°.b. the mean increases by 2.7°. c. the means stays at 80.4°. d. the mean stays at 82.7°.
Answer:
The measure which changes the most is the mean and the new value is 84°
The correct answer option is option B
Step-by-step explanation:
How to calculate mean and median?
Given data set:
58, 61, 71, 77, 91, 100, 105, 102, 95, 82, 66, 57
Median = (100 + 105) / 2
= 205/2
= 102.5°
Mean = (58 + 61 + 71 + 77 + 91 + 100 + 105 + 102 + 95 + 82 + 66 + 57) / 12
= 965 /12
= 80.4°
Range = Highest value - lowest value
= 105 - 57
= 48°
Interquartile range = Third quartile - First quatile
= 88.5° - 74°
= 14.5°
If 58° is changed to 101°
101, 61, 71, 77, 91, 100, 105, 102, 95, 82, 66, 57
Median = (100 + 105) / 2
= 205/2
= 102.5°
Mean = (101 + 61 + 71 + 77 + 91 + 100 + 105 + 102 + 95 + 82 + 66 + 57) / 12
= 1008 /12
= 84°
Range = Highest value - lowest value
= 105 - 57
= 48°
First quartile = (71 + 77) / 2
= 148/2
= 74°
Third quartile = (95 + 82)/2
= 177/2
= 88.5°
Interquartile range = Third quartile - First quatile
= 88.5° - 74°
= 14.5°
Therefore, the new mean is 84°
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pls answerrrrr
with simple working
Answer:
it goes by minus 8 each time so just put the number and negative 8 each time on a paper and the 20th term would be your answer
PLSSSS HELPP WILL GIVE BRAINLIEST!!!!
Answer:
Proved
Step-by-step explanation:
In triangles ABD and CBD,
Angle CDB = Angle ADB (given in Question)
Angle CBD = Angle ABD ( Angle Bisector Therom)
BD = BD ( Common side)
Therefore Triangle ABD is congruent to Triangle CBD by ASA
Hope it helps :)
Can someone please help me with this asap? I will give brainliest if it’s all done correctly
Show work please
Answer:
The probability of selecting a face card or a spade when randomly selecting one card from a standard deck of cards is 11/26 or approximately 0.423.
Step-by-step explanation:
Part A
No, selecting a face card and selecting a spade when randomly selecting one card from a standard deck of cards are not mutually exclusive events. This is because there are face cards that are also spades, so there is an overlap between the two events.
Part B
The probability of selecting a face card or a spade when randomly selecting one card from a standard deck of cards can be calculated as follows
P(face card or spade) = P(face card) + P(spade) - P(face card and spade)
Part C
The probability of selecting a face card or a spade when randomly selecting one card from a standard deck of cards can be calculated as follows
P(face card or spade) = P(face card) + P(spade) - P(face card and spade)
P(face card or spade) = 12/52 + 13/52 - 3/52
P(face card or spade) = 22/52
P(face card or spade) = 11/26
Hence, the probability of selecting a face card or a spade when randomly selecting one card from a standard deck of cards is 11/26 or approximately 0.423.
Okay, I really need help, it’s a cooking need of help so I was making pasta, and I used a smaller top on a big pot (I know, I know..) and now it won’t come off, I’ve tried banging it around the edges, I put cold water on it, and I used dish soap to pry it open, nothing. I am not putting it in the freezer for it to cool off. Please help me!
Answer:
Dry your hands from the soap and try twisting the lid of the pan. This will cause it to start to turn. If you put it in the freezer it will be harder to take it off. My guess is by twisting the lid, you would be able to cause enough friction to take the lid off.
Step-by-step explanation:
If this doesn't work try tilting the lid and then lifting it
Answer:
Try soaking it in hot water. The pot only if possible. The top is smaller and if the pan expands it should loosen up
Find m angle 1 and angle 2
According to the figure and using the corresponding angle theorem we have that
angle 1 = 82 degreesangle 2 = 122 degreesHow to find angle 1 and 2Corresponding angles are pairs of angles that are formed when a straight line intersects two parallel lines.
They are located on opposite sides of the transversal (the line that intersects the parallel lines) and in corresponding positions. Corresponding angles have the same degree measurement as each other.
angle 1 = 82 degrees corresponding angles are equal
angle a = 58 degrees corresponding angles are equal
angle a and angle 2 are linear pair hence angle a and angle 2 are supplementary angles
hence we have that
angle a + angle 2 = 180 degrees
58 + angle 2 = 180
angle 2 = 180 -58
angle 2 = 122 degrees
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Which of the following gives the correct range for the graph?
A coordinate plane with a segment going from the point negative 5 comma negative 2 to 0 comma negative 1 and another segment going from the point 0 comma negative 1 to 2 comma 3.
a −5 ≤ x ≤ 2
b −5 ≤ y ≤ 2
c −2 ≤ x ≤ 3
d −2 ≤ y ≤ 3
Answer: Its B.) −5 ≤ y ≤ 2
Step-by-step explanation: The graph shows one part of the interval increasing which is the y value and stops at 3. The X interval which is the decreasing interval stops at -5. So this would be -5 greater then or equal to y greater then or equal to 2.
Sorry if wrong! Have a nice day :)
w is proportional to d
when d=35 the value of w 133
work out the value of w when d=25
Accοrding tο given proportional data when d = 25, the value οf W is 95.
What is prοpοrtiοnality?Relatiοnships between the twο variables that are prοpοrtiοnal οccur when their ratiοs are equal. Anοther apprοach tο cοnsider is that in a prοpοrtiοnal relatiοnship, οne variable is cοnsistently equal tο the οther's cοnstant value. This parameter is referred tο as the "cοnstant οf prοpοrtiοnality".
Accοrding tο the given infοrmatiοn:
Using the prοpοrtiοnality relatiοn W ∝ d, we can set up the fοllοwing prοpοrtiοn:
W/d = k, where k is the cοnstant οf prοpοrtiοnality.
Tο sοlve fοr k, we can use the values given in the prοblem:
133/35 = k
Simplifying, we get:
k = 3.8
Nοw, we can use this value οf k tο find the value οf W when d = 25:
W/25 = 3.8
Simplifying, we get:
W = 95
Therefοre, accοrding tο given infοrmatiοn when d = 25, the value οf W is 95.
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Please Show detailed algebraic work.
As a result, it takes around 0.25 minutes for the oven to reach 300°F, or equation roughly 15 seconds. Rounded to the closest minute, the answer is 0 minutes.
What is equation?A math equation is a mechanism for connecting statements and indicating equivalence with both the equals symbol (=). An equation is an algebraic claim that asserts the equivalency of two formulae in algebra. In the calculation 3x + 5 = 14, for example, the equivalence sign separates the numbers 3x + 5 and 14. To explain the relationship among the two words on each surface of a letter, a formula can really be employed. The software and the logo are usually interchangeable. For instance, 2x - 4 equals 2.
Set T = 300 in the equation and solve for t to discover the number of minutes it takes for the oven's temperature to reach 300°F:
T = 400 - It
300 = 400 - It / subtraction T = 300
It = 400 - 300
It = 100
t = 100 / I
T = 400 minus It I = (400 minus T) / t
When we substitute T = 0 and T = 1, we get:
I = (400 - 0) / 1\sI = 400
Now we can substitute I = 400 into the equation we found before for t:
t = 100 / I\st = 100 / 400\st = 0.25
As a result, it takes around 0.25 minutes for the oven to reach 300°F, or roughly 15 seconds. Rounded to the closest minute, the answer is 0 minutes.
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Identity the dependent variable and the independent variable in each situation
The distance traveled, d, and the speed, s
Situation 1:
Dependent variable: Distance traveled (d)
Independent variable: Speed (s)
Situation 2:
Dependent variable: Speed (s)
Independent variable: Distance traveled (d)
What are dependent variable?The dependent variable is the variable being studied or measured in an experiment or observation. It is the output or response that is being examined based on changes in the independent variable.
What are independent variable?An independent variable is a variable that is manipulated or controlled by the experimenter in order to observe the effect on the dependent variable. It is also sometimes called the predictor variable.
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Use the function below to find F(4). F(x)=5 times (1/3)^x
Answer: 5/81
Step-by-step explanation:
F(4) = 5 * (1/3) ^ 4
F(4) = 5 * 1/81
F(4) = 5/81
Find the distance, c, between (3, –4) and (–7, 1) on the coordinate plane. Round to the nearest tenth if necessary.
Answer:
c ≈ 11.2
Step-by-step explanation:
Using the distance formula, we have:
c = √[(x2 - x1)² + (y2 - y1)²]
c = √[(-7 - 3)² + (1 - (-4))²]
c = √[(-10)² + 5²]
c = √(100 + 25)
c = √125
c ≈ 11.2
Therefore, the distance between the two points is approximately 11.2 units.
[tex]\sqrt{(-7-3)^{2} + (1-(-4))^{2} } = \sqrt{(-10)^{2} + (5)^{2} } = \sqrt{100 + 25} \\[/tex]
[tex]= \sqrt{125} = 5\sqrt{5} =11.2[/tex]
molly has a $2500 down payment saved for this purchase, and the dealers $1500 cash allowance will come straight off her total. How much loan does molly need?
Based on the given question, Molly needs a loan of $1,000
What is a Loan?A loan occurs when one or more people, businesses, or other entities lend money to other people, businesses, or other entities.
The recipient incurs a debt and is often responsible for both the principal amount borrowed as well as interest payments until the debt is repaid.
How to calculate the amount of loan that Molly needs?
Molly has a $2500 down payment saved for a purchase
The dealer's cash allowance is $1500, this will be deducted from her total
The amount of loan that Molly needs can be calculated by subtracting the dealer's cash allowance from Molly's down payment
Loan= Down payment-cash allowance= 2500 - 1500= 1000
Hence Moly needs a loan of $1,000
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Compare two functions: f(x) shown in the table below and g(x) which is described in the following way: Darya is 2 miles east of home and begins jogging east at a constant rate of 5 miles per hour. Let g(x) be her distance away from home and x be the time she has been jogging in hours.
Which of the following statements is NOT true?
f(x) is an exponential function whose base is 3.
f(x) is an exponential growth function that will keep growing larger.
g(x) is an exponential function because Darya keeps getting farther away from home.
g(x) is a linear function because Darya is jogging at a steady rate.
It is untrue to say that the exponential function f(x) has a base of 3.
What does the function's exponential mean?A mathematical function termed an exponential function has the formula f (x) = axe, where x is a variable and a constant called the base of the function, which must be greater than 0. The most often used exponential function basis is the transcendental number e, or approximately 2.71828.
What is an example of an exponential function?An exponential function has the form f(x) = bx for b > 0 and b 1. Like in any exponential expression, the base is b and the exponent is x. Bacterial growth serves as an example of an exponential function. Some bacteria grow by two per hour.
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What equation is created by this ordered pair (1,1) (2,2) (3,4) (4,7) (5,12)?
The answer will be as follows after answering the supplied question It is equation worth noting that this equation is a cubic polynomial.
What is equation?A math equation is a process that relates two statements by using the equals sign (=) to indicate equivalence. In algebra, an equation is a mathematical statement that shows the equivalence of two mathematical expressions. In the equation 3x + 5 = 14, for example, the equal sign separates the numbers 3x + 5 and 14. A mathematical formula may be used to understand the link between the two phrases written on opposite sides of a letter. Frequently, the logo and the software are the same. For example, 2x - 4 = 2.
The degree n Lagrange polynomial that passes through n+1 unique points (x0,y0), (x1,y1),..., (xn,yn) is given by:
[j=0 to n] L(x) [i=0 to n, ij] * yj (x - xi)/(xj - xi) (xj - xi)
We have 5 points in our situation, thus n = 4. We get the following results by plugging the provided points into the Lagrange polynomial formula:
[tex]L(x) = 1*(x-2)(x-3)(x-4)(x-5)/(-1123) + 2(x-1)(x-3)(x-4)(x-5)/(1123) + 4(x-1)(x-2)(x-4)(x-5)/(1123) + 4(x-1)(x-2)(x-4)(x-5)/(1123) (2123)+ 12(x-1)(x-2)(x-3)(x-4)/(432*1)[/tex]
When we simplify this expression, we get:
[tex]L(x) = x^3/3 - 3x^2/2 + 11x/6 - 1[/tex]
As a result, the equation of the curve passing through the given locations is:
[tex]y = x^3/3 - 3x^2/2 + 11x/6 - 1[/tex]
It is worth noting that this equation is a cubic polynomial.
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Language arts Algebra 1 T.16 Compare linear functions: tables, graphs, and equations GOT Function A Function A and Function B are linear functions. -10-22 9 Analytics -20 16 Function B y = 3x Which statement is true? Science Submit The y-intercept of Function A is greater than the y-intercept of Function 8. & Recommendations The y-intercept of Function A is less than the y-intercept of Function B. Work it out Not feeling rea These help:
The true statements about the functions A and B are
C. The y-intercept of Function A is greater E. The rate of change of Function A is lesserHow to determine the true statement about the functionsGiven that
Function B: y = 3x + 4
A linear equation is represented as
y = mx + c
Where
Rate of change = m
y-intercept = c
Using the above definition, we have the following summary for function B
Rate of change = 3 and y-intercept = 4
For function A (the table), we have
Rate of change = (10 - 2)/(3 - 1)
Rate of change = 4
While the y-intercept is
y-intercept = 2 - 4
y-intercept = -2
So, we have
Function A Function B
Rate of change 3 4
y-intercept 4 -2
By comparing the features, we have the following true statements
C. The y-intercept of Function A is greater than the y-intercept of Function B.
E. The rate of change of Function A is less than the rate of change of Function B.
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Complete question
Function A and Function B are linear functions. Function A is represented by the table of values. Function B is represented by the equation.
Function A
x 1 3 4 7
y 2 10 14 26
Function B
y = 3x + 4
Which statements about the properties of Function A and Function B are true? Select each correct statement.
A. The y-intercept of Function A is equal to the y-intercept of Function B. B. The y-intercept of Function A is less than the y-intercept of Function B. C. The y-intercept of Function A is greater than the y-intercept of Function B.
D. The rate of change of Function A is equal to the rate of change of Function B.
E. The rate of change of Function A is less than the rate of change of Function B.
F. The rate of change of Function A is greater than the rate of change of Function B.
Martin is x years old. Jennifer is 3 years younger than Martin. Connor is twice as old as Martin. Write an expression for Jennifer’s age.
Answer:
x-3
Step-by-step explanation:
martin = x
jennifer = x-3
connor = 2x
what is jennifer's age?
Which inequality is equivalent to 5(2x-6)+6(3-x)<2x+4
By answering the given question, we may state that Therefore, the equivalent inequality is: x < 2
What is inequality?A relationship between two terms or values which is not equal is known as an inequality in mathematics. Inequality follows imbalances, therefore. In mathematics, an inequality makes a link among two quantities that are not equal. Egality and inequality are different. Use the not similar symbol most frequently when two components are not equal (). Various inequalities are employed to contrast values of all sizes. By changing the two parts until only the variables are left, one can solve many straightforward inequalities. However a variety of factors support inequality: Negative values are split or added on either side. Switch between the left & right.
inequality using basic algebraic operations:
5(2x-6)+6(3-x)<2x+4
10x-30+18-6x < 2x+4
4x - 8 < 0
4x < 8
x < 2
Therefore, the equivalent inequality is:
x < 2
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Three dozen blocks were in a bag. Four were red. What is the probability that a block removed at random will NOT be red?
Answer:
9
Step-by-step explanation:
1 dozen -12
there dozen - 36
thus
red blocks -4
36/4
= 9
10 POINTS! PLEASE HELP! CORRECT ANSWERS ONLY!
Answer:5f+6g+12
Step-by-step explanation:
10f-5f=5f
8+4=12
5f+6g+12
You are currently at point A on a circle, and you want to get to point B on the circle. You are trying to minimize the distance walked. You can either walk into the middle of the circle and then out of the circle, or you can walk around the edge of the circle. At what minimum distance between A and B do you walk in and then out of the circle in order to minimize the distance walked? State the answer as a % of total circumference.
To minimize the distance walked, and the minimum distance is 31.8% of the total circumference of the circle.
What is the minimum distance between A and B?
If we draw a line segment from A to B, it will intersect the circle at two points, C and D. We can find the distance AC and the distance BD by using the Pythagorean theorem.
Let r be the radius of the circle, and let x be the distance we walk in and then out of the circle.
Then, we have:
AC = √(r^2 - (x/2)^2)
BD = √(r^2 - (x/2)^2)
The total distance we walk is AC + BD + AB. Since AB is fixed, we want to minimize AC + BD.
AC + BD = 2√(r^2 - (x/2)^2)
To minimize this expression, we need to maximize the value inside the square root. That means, we need to set (x/2)^2 to be as small as possible.
In other words, we want x to be as large as possible.
The maximum value of x is the diameter of the circle, which is 2r.
If we walk in and out of the circle along the diameter, then AC and BD are both equal to r, and the total distance we walk is 2r + AB.
Therefore, the percentage of the total circumference that we walk in and out of the circle is:
2r / (2πr) * 100% = 100/π ≈ 31.8%
So, we need to walk in and out of the circle at two points that are opposite to each other, along a diameter, to minimize the distance walked, and the minimum distance is approximately 31.8% of the total circumference of the circle.
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