(a) gcd(a, b) must equal 1.
(b) No, it is not necessarily true that gcd(a, b) = 6 just because au + bv = 6.
What do you mean by integer?An integer is a whole number that can be positive, negative, or zero. It is a number without any fractional or decimal component. Integers include the counting numbers (1, 2, 3, ...) and their negatives (-1, -2, -3, ...), as well as 0. Examples of integers are -5, -3, -1, 0, 1, 2, 3, 5, and so on. Integers are used in a variety of mathematical operations, such as addition, subtraction, multiplication, and division, and they play a key role in many areas of mathematics, including number theory, algebra, and discrete mathematics.
(a) Suppose that gcd(a, b) is not equal to 1. Then, there exists a positive integer k such that k divides both a and b. Let a = ka' and b = kb', where a' and b' are relatively prime.
Since there are integers u and v satisfying au + bv = 1, it follows that ka'u + kb'v = k(a'u + b'v) = k. But this means that k divides 1, which is a contradiction. Hence, gcd(a, b) must equal 1.
(b) No, it is not necessarily true that gcd(a, b) = 6 just because au + bv = 6. Consider the counter example a = 4 and b = 2. In this case, we have 2u + 4v = 6, but gcd(a, b) = 2, not 6.
In general, if there are integers u and v satisfying au + bv = k, then gcd(a, b) divides k. However, the converse is not necessarily true; that is, just because gcd(a, b) divides k does not mean there are integers u and v satisfying au + bv = k.
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Gcd(a,b) can take on any value from 1 to the minimum of a and b (inclusive). For example, if a = 6 and b = 1, then au + bu = 6, but gcd(6,1) = 1.
What is GCD?GCD is an abbreviation for Greatest Common Divisor. It is the greatest positive integer that divides two or more numbers without a remainder. It is used to simplify fractions and solve linear Diophantine equations. The Euclidean algorithm or the prime factorization method can be used to compute GCD.
(a) Assume u and v are integers such that au + bv = 1. So a and b must have no common factor, because the only way to add two integers and obtain 1 is if they have no common factors. As a result, gcd(a,b) = 1.
(b) No, it is not always the case that gcd(a,b) = 6. For example, if an is 6 and b is 1, au + bu equals 6, while gcd(6,1) equals 1. In general, gcd(a,b) can have any value between 1 and the minimum of a and b. (inclusive).
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The left and right page numbers of an open book are 2 consecutive integers whose sum is 295 what are the page numbers
The page numbers of the book are 147 and 148.
What are consecutive numbers?Consecutive numbers are numbers that follow each other in order. They have a difference of 1 between every two numbers.
In a set of consecutive numbers,the mean and the median are Equal. If n is a number, then n, n+1, and n+2 would be consecutive numbers.
Given,
Left and right page numbers of a book are consecutive integers.
Sum of both integers = 295
Let x is left integer then right integer is x + 1
Equation becomes
x + x + 1 = 295
2x + 1 = 295
2x = 295 - 1
x = 294/2
x = 147
Left integer = 147,
Right integer = 147 + 1 = 148
Hence, 147 and 148 are the page numbers of the book.
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Determine the sum of the first 4 terms of "0.981 using binomial expansion. Foot Cab (5 marks) (5 marks)
The sum of the first 4 terms of the binomial expansion of (0.9 + 0.081)³ is approximately 0.9441.
What is binomial expansion?The binomial expansion formula is given by:
[tex](a + b)^n = nC_0 \times a^n \times b^0 + nC_1 \times a^{(n-1)} \times b^1 + nC_2 \times a^{(n-2)} \times b^2 + ... + nC_n \times a^0 \times b^n[/tex]
where nCk is the binomial coefficient, which is equal to n! / (k! * (n-k)!) for k = 0, 1, 2, ..., n.
Here, we have a = 0.9 and b = 0.081, and we want to find the sum of the first 4 terms, which means we need to find:
(0.9 + 0.081)⁰ + (0.9 + 0.081)¹ + (0.9 + 0.081)² + (0.9 + 0.081)³
Using the binomial expansion formula with n = 3, we get:
(0.9 + 0.081)³ = 3C0 x 0.9³ x 0.081^0 + 3C1 x 0.9² x 0.081¹ + 3C2 x 0.9¹ x 0.081² + 3C3 x 0.9⁰ x 0.081³
Simplifying each term, we get:
1 x 0.729 x 1 + 3 x 0.81 x 0.081 + 3 x 0.9 x 0.006561 + 1 x 0.000531441
= 0.729 + 0.19683 + 0.0177117 + 0.000531441
= 0.944076141
Therefore, the sum of the first 4 terms of the binomial expansion of (0.9 + 0.081)³ is approximately 0.9441 (rounded to 4 decimal places).
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please help this is due today :(
Answer:
C
Step-by-step explanation:
Supplementary angles are two angles that equal 180 degrees. An angle is 180 degrees when it's a straight line. As you see, the line with E, A, P, and B is a straight line, making A and B supplementary.
Answer: Supplementary angles because the 2 angles when they are joined they both add up to 180 degrees
Step-by-step explanation:
It was recently estimated that domestic vehicles outnumber foreign vehicles by about seven to five. If there are 4152 vehicles in a county, how many of them are domestic?
In a case whereby It was recently estimated that domestic vehicles out number foreign vehicles by about seven to five. If there are 4152 vehicles in a county, the number of them that are domestic is 2422 domestic vehicles.
How can the number of domestic vehicle be determined?The concept that will be used here is simplification.
From the question, we were told that there are 7x domestic vehicles as well as 5x foreign vehicle.
We can see that the x is the unknown number, which we need to find out,
Then we can make an equation as:
7x + 5x = 4152
12x= 4152
x=4152/12
=346
Hence we can say that there are 7*346 = 2422 domestic vehicles
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Solve with the box method
Using the box method, the expression is solved as: 4x³ + 8y² - 19y - 35 (see image below).
How to Expand an Algebraic Expression Using the Box Method?The box method, also known as the grid method or area model, is a visual technique for multiplying two expressions or expanding algebraic expressions. Here are the steps to expand an algebraic expression using the box method:
Write the expression to be expanded: Write the algebraic expression that you want to expand.Draw a box: Draw a box, and divide it into smaller boxes, one for each term in the expression.Write the terms: Write each term of the expression on the top of each box.Multiply the terms: Multiply the terms along the sides of the boxes, and write the result in the corresponding smaller box.Combine like terms: Add the products in each box to obtain the expanded expression.The box method shows how the expression has been expanded, thus, combining like terms, we have:
4x³ + 8y² - 19y - 35
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A cylindrical tank has a radius of 20 cm. Water flows out of the tank at .05 liters per second. How fast does the level of water in the tank drop?
Answer:
Step-by-step explanation:
To calculate the rate at which the water level drops, we need to find the volume of water being lost per second and convert that to a height change. The volume of water being lost is given as 0.05 liters per second. The volume of a cylinder with height h and radius r is given by the formula:
V = πr^2h
We can rearrange this formula to solve for h:
h = V / (πr^2)
Substituting the given values, we get:
h = 0.05 / (π * 20^2) = 0.05 / (π * 400) = 0.05 / 1257.0796
So the rate at which the water level is dropping is approximately 0.04 cm per second.
Similar triangles: To indirectly measure the distance across a river, Madeline stands on one side of the river and uses sight-lines to a landmark on the opposite bank. Madeline draws the diagram below to show the lengths and angles that she measured. Find PR, the distance across the river. Round your answer to the nearest foot.
Answer:
PR = 288.75 feet
Step-by-step explanation:
There are two triangles that we need to consider:
ΔPRE and ΔPOC
These triangles are similar since
m∠PRE = m∠POC = 90°
∠PRE ≅ ∠POC
∠RPE ≅ ∠OPC (common angle)
AA Similarity theorem..
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
∴ ΔPRE ≅ ΔPOC
If two triangles are similar, the ratios of the corresponding sides must be equal
⇒ [tex]\dfrac{PO}{PR} = \dfrac{OC}{PE}\\\\[/tex] (1)
PO = PR + RO
PR is the width of the river that is to be determined
Let PR = x
Then
PO = x + 165
OC = 330, RE = 210
[tex]\dfrac{PO}{PR} = \dfrac{OC}{PE}\\\\\Rightarrow \dfrac{x + 165}{165} = \dfrac{330}{210}\\\\[/tex]
Left side:
[tex]\dfrac{x+165}{x} = \dfrac{x}{x} + \dfrac{165}{x} = 1 + \dfrac{165}{x}[/tex]
Simplify [tex]\dfrac{330}{210}[/tex] by dividing both numerator and denominator by 30
=>
[tex]\dfrac{330}{210} = \dfrac{11}{7}\\\\[/tex]
Therefore
[tex]\dfrac{PO}{PR} = \dfrac{OC}{PE}\\\\\Rightarrow 1 + \dfrac{165}{x} = \dfrac{11}{7}\\[/tex]
Subtract 1 from both sides:
[tex]\dfrac{165}{x} = \dfrac{11}{7} - 1\\\\\dfrac{165}{x} = \dfrac{4}{7}\\\\[/tex]
Cross-multiply
[tex]165 \times 7 = 4x\\\\\Rightarrow 4x = 165 \times 7\\\\\Rightarrow x = \dfrac{165 \times 7 }{4} = 288.75\;feet[/tex]
PLEASE HELP!
Determine the terms and the coefficients of the terms for the
polynomial
x^2-3x+2
Complete the table for the polynomial
Answers in bold
[tex]\begin{array}{|c|c|} \cline{1-2}\text{Term} & \text{Coefficient}\\\cline{1-2}\text{x}^2 & \boldsymbol{1}\\\cline{1-2}\boldsymbol{-3\textbf{x}} & -3\\\cline{1-2}2 & \boldsymbol{2}\\\cline{1-2}\end{array}[/tex]
Explanation:
The given expression [tex]\text{x}^2-3\text{x}+2[/tex] is the same as [tex]\text{x}^2+(-3\text{x})+2[/tex]
This leads to the three terms as shown in the table above. Each term is separated by a plus sign.
The coefficient is the number to the left of the variable. Think of [tex]\text{x}^2[/tex] as [tex]1\text{x}^2[/tex] and think of [tex]2[/tex] as [tex]2\text{x}^0[/tex]
In other words, [tex]\text{x}^2-3\text{x}+2[/tex] is the same as [tex]1\text{x}^2+(-3\text{x})+2\text{x}^0[/tex]
I checked the answer for this question and its. Angle PSR = ⁴⁄₇x = ⁴⁄₇ x 105 = 60°
What I don't understand is how they find the 105°. Could someone write down a step by step explanation answer to this question please?
Answer:
see explanation
Step-by-step explanation:
(a)
let ∠ QRS = x then ∠ PQR = x and
∠ QPS = [tex]\frac{6}{7}[/tex] x and ∠ PSR = [tex]\frac{4}{7}[/tex] x
the sum of the 4 angles in a quadrilateral = 360°
sum the 4 angles and equate to 360
x + x + [tex]\frac{6}{7}[/tex] x + [tex]\frac{4}{7}[/tex] x = 360 ( multiply through by 7 to clear the fractions )
7x + 7x + 6x + 4x = 2520
24x = 2520 ( divide both sides by 24 )
x = 105
then
∠ QRS = 105° , so
∠ PSR = [tex]\frac{4}{7}[/tex] × 105° = 4 × 15° = 60°
(b)
∠ QPS = [tex]\frac{6}{7}[/tex] × 105° = 6 × 15° = 90° ← a right angle
What are the terms a0, a1, a2 and a3 of the sequence {an}, where an equals
a. 2n + n
b. n(n+1)!
c. [n/2]
d. [n/2] + [n/2]
The terms a0, a1, a2, and a3 of the sequence {an} in each case are as follows:
a. a0 = 0, a1 = 3, a2 = 6, a3 = 9
b. a0 = 0, a1 = 2, a2 = 12, a3 = 72
c. a0 = 0, a1 = 0.5, a2 = 1, a3 = 1.5
d. a0 = 0, a1 = 1, a2 = 2, a3 = 3
The sequence {an} is defined asa. 2n + n
a0 = 2*0 + 0 = 0
a1 = 2*1 + 1 = 3
a2 = 2*2 + 2 = 6
a3 = 2*3 + 3 = 9.
b. n(n+1)!
a0 = 0(0+1)! = 0
a1 = 1(1+1)! = 2
a2 = 2*(2+1)! = 12
a3 = 3(3+1)! = 72
c. [n/2]
a0 = [0/2] = 0
a1 = [1/2] = 0.5
a2 = [2/2] = 1
a3 = [3/2] = 1.5
d. [n/2] + [n/2]
a0 = [0/2] + [0/2] = 0
a1 = [1/2] + [1/2] = 1
a2 = [2/2] + [2/2] = 2
a3 = [3/2] + [3/2] = 3
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A random sample of 190 students were asked what kind of vehicle they prefer- a car or truck. The
following contingency table gives the two-way classification of the responses.
P(Female) = 111, P(Car) = 132, P(Female | truck) = 19 P(Truck | female)
What is probability?
Probability is a branch of math which deals with finding out the likelihood of the occurrence of an event. Probability measures the chance of an event happening.
From the image in the question,
P(Female) = 92 + 19
= 111
P(Car) = 40 + 92
= 132
P(Female | truck) = 19
P(Truck | female) = 19
The events "Female" and "Truck" are dependent
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In a class of 30 children, 6 of the children are girls. What percentage of the class are girls?
Answer: 20% of the class is girls.
6/30 = 0.2
0.2=20%
find the sizes of the lettered angle
[tex]find \: the \: sizes \: of \: the \: lettered \: angle[/tex]
The measures of the angles are:
n = q = j = 57°
m = l = p = k = 123°
How to find the measures of the angles?Remember that vertical angles have the same measure, then:
n = 57°
And if the two horizontal lines are parallel, then the values in the correspondent quadrants are the same ones, then:
q = 57°
j = 57°
And adjacent angles add up to 180°, them:
57° + m = 180°
m = 180° - 57°
m = 123°
Then:
m = l = p = k = 123°
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In 2009, a laptop computer was purchased for $800. Each year since, the resale value has decreased by 27%.
Let t be the number of years since 2009. Let y be the value of the laptop computer, in dollars.
Write an exponential function showing the relationship between y and t.
On solving the provided question we can say that by percentage y = $1900(77/100)t
What is percentage?A percentage in mathematics is a figure or ratio that is stated as a fraction of 100. The abbreviations "pct.," "pct," and "pc" are also occasionally used. It is frequently denoted using the percent symbol "%," though. The amount of percentages has no dimensions. With a denominator of 100, percentages are basically fractions. To show that a number is a percentage, place a percent symbol (%) next to it. For instance, if you correctly answer 75 out of 100 questions on a test (75/100), you receive a 75%. To compute percentages, divide the amount by the total and multiply the result by 100. The percentage is calculated using the formula (value/total) x 100%.
In 2005, a laptop computer was purchased for 81900. Each year since, the resale value has decreased by 23%. Let t be the number of years since Let Y be the value of the laptop computer, in dollars. Write an exponential function showing the relationship between Y and
t. value of laptop in 2005 81900 decrease in resale \/aIUe each year
23 77% resale value of laptop after each year = 100
resale value after one x $1,900
Now, Hence relation between value of the laptop computer and time is
y = $1900(77/100)t
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With her new company, Annika has a two-year assignment living in a large city. The company will reimburse most of her expenses. Annika is using the Quantitative Reasoning Process to build a budget. Calculate Annika's estimated living expenses for the next two years using the following:
Rent: $1,050 per month, Gym membership: $325 per year with a monthly fee of $35, Renters' insurance: $70 every 6 months, Groceries: $300 per month, and Eating out & entertaining: $150 per week.
Assume 52 weeks in 1 year and 12 months in 1 year.
(Round your answer to the nearest whole dollar)
The solution is, Total estimated living expenses = $53,900.
What is multiplication?In mathematics, multiplication is a method of finding the product of two or more numbers. It is one of the basic arithmetic operations, that we use in everyday life.
here, we have,
Estimated expenses for next 2 years:
Rent = 1100 * 12 * 2 = 26,400
Gym membership = 450 * 2 = 900
Gym monthly fee = 40 * 12 * 2 = 960
Renter's insurance = 60 * (12 * 2 / 6) = 240
Groceries = 300 * 12 * 2 = 7200
Eating out & entertainment = 175 * 52 * 2 = 18,200
Total estimated living expenses = $53,900
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Please indicate which one is the minimal complete set of relational algebra operations in the following: a. {0, 1, U, -, 1} b. {0, T, U, n, x} c. {0, 1, 0, -, *}d. {C, T, U, -, *}
D i.e. {C, T, U, -, *} is the minimal complete set of relational algebra operations.
What is relational algebra?
Relational Algebra is a query language that takes a Relation as input and returns another Relation as output. Relational algebra provides the theoretical basis for relational databases and SQL. The necessary column data for a relation is projected via projection.
Now,
As Five basic operations in relational algebra are : Selection, Projection, Cartesian product, Union, and Set Difference and all these are present in
{C, T, U, -, *}.
Hence,
D i.e. {C, T, U, -, *} is the minimal complete set of relational algebra operations.
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Find the value of X.
The value of the side x is 6. Option D
How to determine the value of the variableIt is important that we know the six different trigonometric identities in mathematics.
These identities are;
sinecosinetangentcotangentsecantcosecantWe have the sides are;
Hypotenuse side = x
Opposite side = y
Angle = 30
Adjacent side = 3
Then,
cos 60 = 3/x
Find the value
x = 6
Hence, the value is 6
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I need to find the slope of the graph at x= -1
And the answer for A, B, C, D (I really need help)
Answer:
See below
Step-by-step explanation:
The derivative function f' is the slope of the function at any given 'x'
A) f ' = 8 x^3 - 24 x^2
B ) put in x = -1 and compute 8 ( -1)^3 -24(-1)^2 = -8 - 24 = - 32 slope
C) we know the slope (-32) and the x coordinate (-1) put this value of x into the original equation to find y = 2(1^4) - 8(-1)^3 + 6 = 16
so point slope form of the line is
y - 16 = -32 (x - -1) re-arrange:
y = -32x -16
D) where slope (f ') = 0 the slope is 0 and thus the tangent line is horizontal 8x^3 - 24x^2 = 0
8x^3 = 24x^2
8x = 24
x = 3
Can someone answer my last questions please.
They are worth 30+ points!!
Answer:
ok
Step-by-step explanation:
later
QUICK!! 50 POINTS PLUS BRAINLIEST AND THANKS
Which programs were parts of FDR's New Deal?
Choose all answers that are correct.
Responses
Works Progress Administration
Civil Rights Act
Federal Deposit Insurance Corporation
Fair Labor Standards Act
Federal Deposit Insurance Corporation, programs were parts of FDR's New Deal.
What is Social Security?Social security is the protection that a society provides to individuals and households to ensure access to health care and to guarantee income security, particularly in cases of old age, unemployment, sickness, invalidity, work injury, maternity or loss of a breadwinner.
here, we have,
One of Franklin D. Roosevelt's most well-known and widespread programmers, Social Security was first introduced in 1935. One of the New Deal initiatives that helped the nation emerge from the Great Depression and recover its assets and wealth as well as give the economy new life was Social Security.
In order to help the aged and disabled better support themselves while they were unable to work, Social Security established safety net protections.
The Social Security programmed and Fannie Mae, or The Federal National Mortgage Association, both still exist today, despite the fact that the majority of the New Deal policies were abandoned when the United States entered World War II. After World War II, Fannie Mae contributed to funding and fueling the housing boom for soldiers coming home.
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Find the inverse of f(x).
Microeconomics
The total cost function of a monopolist is () = 3. The inverse demandfunction for
the monopolist’s products is () = 12 − .
A. find the profit-maximizing level of price and output
B. Find the maximum profit of the firm at an equilibrium price level.
Answer:
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4/3+z = 6/5 Answer please
Answer:
z=0.13 or z=-2/15
Step-by-step explanation:
First you must simply the fraction into a decimal, leaving us with 1.2 and 1.3. Then you subtract 1.3 from 1.2. Leaving us with z=0.13 or in fraction form,
z= -2/15
of all the 100 items i buy today, id like 8% of them to be cupcakes. How many cupcakes is that
Answer:8
Step-by-step explanation:
100*0.08
you would buy 8 cupcakes
Helpppppppp meeee please
Answer:
(a) 44%
(b) 32%
Step-by-step explanation:
(a) In order to find the percentage of students that prefer comedies, we can add up divide the sum of the total number of students who prefer comedies (Twice/month or less + Thrice/month or more) by the total number of students.
Then, we can multiply this number by 100 to find the percentage:
[tex](\frac{45+21}{150})*100\\ (\frac{66}{150})*100\\ 44[/tex]
(b) To find the percentage of students who visit the movies three times a month or more, we can divide the sum of the total number of students who visit the movies three times a month or more by the total number of students.
Like before, we can multiply this number by 100 to find the percentage:
[tex](\frac{21+27}{150})*100 \\(\frac{48}{150})*100\\ 32[/tex]
given an integer variable timer, write a statement that uses the auto-decrement operator to decrease the value of that variable by 1.
timer--; //This statement uses the auto-decrement operator to decrease the value of the variable timer by 1.
The auto-decrement operator -- is used to decrease the value of a variable by 1. In this case, the variable is called timer. The statement which uses the auto-decrement operator is timer--;. This statement will take the current value of the timer variable and decrease it by 1. For example, if the current value of timer is 5, the statement timer--; will decrease the value to 4. This is a very useful and efficient way to decrease a variable's value by 1. It is also much shorter than having to write out a statement such as timer = timer - 1;. The auto-decrement operator can be used with other types of variables as well, such as strings or characters.
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10. From a pack of 2 kg, 40 g of tea was taken. What is the amount of tea left.
Answer:
1960g remaining
Step-by-step explanation:
2kg=total amount of tea
40g=amount of tea taken
Thus, covert 2kg to gram
2×1000=2000g
amount of tea remaining=total amount of tea –amount of tea taken
x=2000–40
x=1960g remaining
Which objective function models the total profit from neck-
laces and wristbands?
Let x = the number of necklaces Mack makes.
Let y = the number of wristbands Mack makes.
A P(x, y) = 25x+40y
B P(x, y)=40x + 25y
c P(x, y) = 3x + 2y
D P(x, y) = 2x + 3y
Zhong has money to invest in one of two accounts for 2
years.
Account 1 requires a $1,200
investment and earns 2.7%
interest compounded daily.
Account 2 requires a $1,000
investment and earns 3.9%
interest compounded annually.
Which account has a higher return?
Answer: We can use the formula for compound interest to calculate the total amount in each account after 2 years.
Account 1: A = P * (1 + r/n)^(nt)
where P = $1,200, r = 2.7%, n = 365 (days/year), t = 2 years.
Substituting the values into the formula, we get:
A = $1,200 * (1 + 0.027/365)^(365 * 2) = $1,200 * (1.0007365)^(365 * 2)
Account 2: A = P * (1 + r)^t
where P = $1,000, r = 3.9%, t = 2 years.
Substituting the values into the formula, we get:
A = $1,000 * (1 + 0.039)^2 = $1,000 * (1.039)^2
Comparing the two amounts, we see that Account 1 has a higher return. The final amount in Account 1 is $1,268.27 and the final amount in Account 2 is $1,079.61.
Step-by-step explanation:
Robert is going on a 2-day canoeing trip with his outdoor adventure group. Robert paddled at a constant sped and travels 3/4 of a mile in 15 minutes.
a. Create a table to determine how long it takes Robert to canoe 1 mile.
With a constant speed Robert took 20 minutes paddle.
What is the speed?The speed formula can be defined as the rate at which an object covers some distance. Speed can be measured as the distance travelled by a body in a given period of time. The SI unit of speed is m/s.
Given that, Robert paddled at a constant speed and travels 3/4 of a mile in 15 minutes.
We know that, Speed = Distance/Time
Speed = 3/4 ÷15/1
= 3/4 × 1/15
= 1/20 miles per minutes
Now, distance = 1 mile and speed =1/20 miles per minutes
1/20 = 1/Time
Time = 20 minutes
Therefore, with a constant speed Robert took 20 minutes paddle.
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