The adjusted balance, considering the new account balance, payments/credits, new purchases, and finance charge, is $489.61.
To calculate the adjusted balance, we need to consider the new account balance, payments/credits, new purchases, and finance charges.
Starting with the new account balance of $435.92, we subtract the payments/credits of $68.50. This represents the amount that has been paid or credited to the account, reducing the balance.
Next, we add the new purchases of $118.49. These are additional charges made to the account, increasing the balance.
Finally, we add the finance charge of $3.70. This charge is typically applied as interest on the outstanding balance.
To calculate the adjusted balance, we can follow these steps:
Start with the new account balance: $435.92
Subtract the payments/credits: $435.92 - $68.50 = $367.42
Add the new purchases: $367.42 + $118.49 = $485.91
Add the finance charge: $485.91 + $3.70 = $489.61
Therefore, the amount of the adjusted balance is $489.61.
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the image is the question
a) c = 22 feet
b) c = 23
c) c = 24
d) c = 30
The length of the triangle's hypotenuse (c) is approximately 22 feet. The closest option provided is "a) c = 22 feet."
The Pythagorean theorem, which asserts that given a right triangle, the sum of the squares of the two shorter sides (a and b), is equal to the square of the hypotenuse (c), can be used to determine the length of the triangle's hypotenuse (c).
a = 10 feet
b = 20 feet
Using the Pythagorean theorem, we can calculate c as follows:
c^2 = a^2 + b^2
c^2 = 10^2 + 20^2
c^2 = 100 + 400
c^2 = 500
To find c, we take the square root of both sides:
c = √500
c ≈ 22.36
Rounding the answer to the nearest whole number, we get c ≈ 22.
Therefore, the length of the triangle's hypotenuse (c) is approximately 22 feet. The closest option provided is "a) c = 22 feet."
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Solve - the mean age of a family of seven is 23 years the median is 16 years the modes are 12 years and 45 years and the range is 35 years. Find the ages of the seven family members.
The ages of the seven family members are 12, 16, 16, 45, 45, 45, and 80 years.
To solve this problem, let's break it down step by step:
1. We are given that the mean age of the family is 23 years. The mean is calculated by summing up all the ages and dividing by the number of family members. Since there are seven family members, the total sum of their ages is 7 * 23 = 161 years.
2. The median age is 16 years. This means that when the ages are arranged in ascending order, the fourth age is 16. Since there are seven family members, the fourth age is the middle age. Therefore, the ages in ascending order are: _ _ 12 16 _ 45 _.
3. The modes are 12 years and 45 years, which means these two ages occur more frequently than any other age. Since the median is 16, it can't be one of the modes. Hence, we can conclude that the family members' ages are: _ _ 12 16 16 45 _.
4. The range is 35 years, which is the difference between the highest and lowest ages. Since the ages are arranged in ascending order, the highest age must be 45 + 35 = 80 years. Therefore, the ages of the family members are: _ _ 12 16 16 45 80.
In summary, the ages of the seven family members are 12, 16, 16, 45, 45, 45, and 80 years.
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9
Type the correct answer in the box. Use numerals instead of words. If necessary, use / for the fraction bar(s).
A system of linear equations is given by the tables. One of the tables is represented by the equation y = -x + 7
y
9
8
X
0
3
6
9
y
5
6
7
8
X
-6
-3
0
3
7
6
The equation that represents the other equation is y= 1/3
The solution of the system is (
)
X+
Reset
5
Next I
4
Number of Years
m
N
1
16
18
18°
19°
22°
30°
20
Mark this and return
28
22
24
26
Average Daily Temperature
30
The mean of the temperatures in the chart is 24° with a standard deviation of 4°. Which temperature is within one
standard deviation of the mean?
32
Save and Exit
Next
Submit
The temperature of 30° is within one standard deviation of the mean.
To determine which temperature is within one standard deviation of the mean, we need to consider the range that falls within one standard deviation above and below the mean.
Given that the mean temperature is 24° with a standard deviation of 4°, one standard deviation above the mean would be 24° + 4° = 28°, and one standard deviation below the mean would be 24° - 4° = 20°.
Looking at the temperatures in the chart, we can see that the temperature of 30° is within one standard deviation of the mean. It falls within the range of 28° (one standard deviation above the mean) and 20° (one standard deviation below the mean).
Therefore, the temperature of 30° is within one standard deviation of the mean.
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The table shows the daily high temperature (°F) and the number of hot chocolates sold at a coffee shop for eight randomly selected days.
The line of best fit for the data in this problem is given as follows:
y = -0.5x + 60.
How to define a linear function?The slope-intercept equation for a linear function is presented as follows:
y = mx + b
In which:
m is the slope.b is the intercept.Two points on the scatter plot are given as follows:
(30, 45) and (60, 30).
When x increases by 30, y decays by 15, hence the slope m is given as follows:
m = -15/30
m = -0.5.
Hence:
y = -0.5x + b.
When x = 30, y = 45, hence the intercept b is obtained as follows:
45 = -15 + b
b = 60.
Thus the function is given as follows:
y = -0.5x + 60.
Missing InformationThe data is given by the image presented at the end of the answer, and the problem asks for the line of best fit for the data.
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An architect is designing a swimming pool with a base in the shape of a right triangle according to the architect the pools depth should be 6 feet less than It’s length x and it’s width should be 8 feet less than it’s length the volume of water in the pool cannot exceed 1680 cubic feet which statement
Which of the following functions is graphed below?
Answer: C
Step-by-step explanation:
the x - 2 means go right 2
the +3 means go up 3
Need help solving the problem, please.
The equation y = -6x + 2 (option c) is parallel to the graph of y = -6x + 3.
Which of the given lines is parallel to y = -6x + 3?The slope-intercept form is expressed as;
y = mx + b
Where m is slope and b is the y-intercept.
Given the equation of the graph in the question:
y = -6x + 3
To determine which of the given options:
a) y = (1/6)x + 3
b) y = -(1/6) + 3
c) y = -6x + 2
d) y = 3x - 6
is parallel to the graph of y = -6x + 3, we need to compare their slopes.
The given equation of the graph is y = -6x + 3:
Slope of the graph is -6.
Now, lets check each option:
a) y = (1/6)x + 3
This equation has a slope of 1/6, which is not equal to -6.
Therefore, it is not parallel to y = -6x + 3.
b) y = -(1/6) + 3
This equation also has a slope of 1/6 (the negative sign doesn't affect the slope), it is not parallel to y = -6x + 3.
c) y = -6x + 2
This equation has a slope of -6, which is the same as the slope of y = -6x + 3. Therefore, it is parallel to the given graph.
d) y = 3x - 6
This equation has a slope of 3, which is not equal to -6. Thus, it is not parallel to y = -6x + 3.
Therefore option C) y = -6x + 2 is the correct answer.
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A spinner has five sectors of equal size the sectors are labeled 1, 2, 3, 4 and five the spinner is spun twice X is the number of times three is fun drag each bar on the horizontal axis to the correct location to create a probability distribution for X
The probability distribution for X is given as follows:
P(X = 0) = 0.64.P(X = 1) = 0.32.P(X = 2) = 0.04.How to calculate a probability?The parameters that are needed to calculate a probability are listed as follows:
Number of desired outcomes in the context of a problem or experiment.Number of total outcomes in the context of a problem or experiment.Then the probability is calculated as the division of the number of desired outcomes by the number of total outcomes.
The distribution gives the probability of each possible outcome, hence it is given as follows:
P(X = 0) = 0.64.P(X = 1) = 0.32.P(X = 2) = 0.04.Learn more about the concept of probability at https://brainly.com/question/24756209
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NO LINKS!! URGENT HELP PLEASE!!!
4. What is a regular polygon?
5. For a regular pentagon, (NOT MULTIPLE CHOICE),
a. Find the measure of a single interior angle.
b. Find the measure of a single exterior angle.
6. The measure of the interior angle of a regular polygon is 162°. How many sides does it have?
Answer:
4.
A regular polygon is a polygon in which all sides are equal in length and all angles are equal in measure.
5.
a. The measure of a single interior angle in a regular pentagon is:
[(n – 2)*180°]/n = 540°/5 = 108°.
b. The measure of a single exterior angle in a regular pentagon is:
360°/n = 360°/5 = 72°.
6.
This can be found using the following formula:
[(n – 2)*180°]/n = Interior angle
(n-2)*180=162°*n
180n-360=162n
180n-162n=360
18n=360
n=360/18
n=20
where n is the number of sides in the regular polygon.
A regular polygon with an interior angle of 162° has 20 sides.
A particular type of vaccine comes in a Brand-1 and a Brand-2. Sixty-five percent
of all patients at a certain vaccination centre want the Brand-2.
i) Among ten randomly selected patients who want this type of vaccine, what
is the probability that at least six want the Brand-2?
ii) Among ten randomly selected patients, what is the probability that the
number who want the Brand-2 vaccine is within 1 standard deviation of
the mean value?
iii) The store currently has seven vaccines of each brand. What is the probability that all of the next ten patients who want this vaccine can get the brand
of vaccine they want from current stock?
i) Probability of at least six patients wanting Brand-2: P(X ≥ 6)
ii) Probability of number of patients within 1 standard deviation of mean: P(μ - σ ≤ X ≤ μ + σ)
iii) Probability that all ten patients get their desired brand from current stock: (7/14) * (6/13) * ... * (1/5)
i) The probability of at least six patients wanting Brand-2 out of ten randomly selected patients can be calculated using the binomial distribution. We need to sum the probabilities of six, seven, eight, nine, and ten patients wanting Brand-2.
The probability can be calculated as P(X ≥ 6) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10), where X follows a binomial distribution with n = 10 and p = 0.65. The answer is the sum of these individual probabilities.
ii) To calculate the probability of the number of patients who want the Brand-2 vaccine being within 1 standard deviation of the mean value, we need to find the range of values that fall within one standard deviation of the mean.
We can use the normal approximation to the binomial distribution since n = 10 is reasonably large. We calculate the mean (μ) and standard deviation (σ) using μ = n * p and σ = √(n * p * (1 - p)), where p = 0.65. Then we calculate the probability of the number of patients falling within the range μ - σ to μ + σ.
iii) Since there are seven vaccines of each brand in stock, the probability that all ten patients who want the vaccine can get the brand they want from the current stock is equal to the probability of the first patient getting their desired brand (7/14) multiplied by the probability of the second patient getting their desired brand (6/13), and so on until the tenth patient (1/5). The final probability is the product of these individual probabilities.
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What is the graph of the solution to the following compound inequality?
3-x22 or 4x+2210
O A.
B.
O c.
O D.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
H
He
+++
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
5 6 7 8 9 10
1
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3
4
€1
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2
5 6 7 8 9 10
3 4 5 6 7 8 9 10
Answer:
Step-by-step explanation:
To graph the solution to the compound inequality 3 - x < 22 or 4x + 2 > 10, we need to graph the individual inequalities and find the overlapping region.
First, let's graph the inequality 3 - x < 22:
Subtract 3 from both sides to isolate x:
-x < 19
Multiply both sides by -1, which reverses the inequality direction:
x > -19
This means that x is greater than -19, but not including -19. So, we will have an open circle at -19 and shade everything to the right of it.
Next, let's graph the inequality 4x + 2 > 10:
Subtract 2 from both sides to isolate 4x:
4x > 8
Divide both sides by 4:
x > 2
This means that x is greater than 2, but not including 2. So, we will have an open circle at 2 and shade everything to the right of it.
Combining the two inequalities, we need to find the overlapping region. Since both inequalities have an open circle at their endpoint, we will use a dashed line to represent them.
The graph should look like this:
markdown
Copy code
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
+ +
| |
| |
+---------------------------------|-------------------->
-19 2
The shaded region will be to the right of -19 and to the right of 2, including all numbers greater than those values.
Therefore, the correct answer is:
O A. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
anwser it pls aaaaaaaassaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
Answer:
Step-by-step explanation:
Volume = Bh
Turn the shape so the trapezoid is on the bottom/base
h=18 for overall shape
B = area of base, trapezoid
B = 1/2 (b₁ + b₂) h
b₁ = 11
b₂ = 25
h = 24 for trapezoid
B = 1/2 (11 + 25)(24)
B = 432
V = Bh
V = (432)(18)
V= 7776 in³
NO LINKS!! URGENT HELP PLEASE!!
Find the value of x
Answer:
x = 4
Step-by-step explanation:
You want the value of x in the figure of a circle with intersecting secants.
Secant relationThe product of lengths from the near and far circle intercepts to the point where the secants intersect is the same for both secants:
6(6+10) = 8(8+x)
6·16 = 8·(8+x)
12 = 8 +x . . . . . . . divide by 8
4 = x . . . . . . . . . . subtract 8
The length x is 4 units.
<95141404393>
y'=y +8z +e^x
x'=2y+z+e^-3x
Answer:
I have not comed across this question before
Simplify 15a6 bc4/ 35a2 c4
The simplified value of the expression given is 3a⁴b/ 7
Given the fraction :
15a⁶bc⁴/ 35a²c⁴divide the coefficients by 5
3a⁶bc⁴/ 7a²c⁴From division rule of indices, subtract the powers of values with Equivalent coefficients.
Hence,
coefficient of of a = 6-2 = 4coefficient of c = 4-4 = 0coefficient of b = bFinally we have :
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CAPM Elements
Value
Risk-free rate (rRF
)
Market risk premium (RPM
)
Happy Corp. stock’s beta
Required rate of return on Happy Corp. stock
An analyst believes that inflation is going to increase by 2.0% over the next year, while the market risk premium will be unchanged. The analyst uses the Capital Asset Pricing Model (CAPM). The following graph plots the current SML.
Calculate Happy Corp.’s new required return. Then, on the graph, use the green points (rectangle symbols) to plot the new SML suggested by this analyst’s prediction.
Happy Corp.’s new required rate of return is .
The new required rate of return for Happy Corp. can be calculated using the Capital Asset Pricing Model (CAPM). The formula for CAPM is:
Required rate of return = Risk-free rate + Beta * Market risk premium
Since the analyst believes that the market risk premium will be unchanged, the only factor that will affect the new required return is the risk-free rate.
Given that the analyst predicts a 2.0% increase in inflation, the risk-free rate will also increase by that amount. Therefore, the new required rate of return for Happy Corp. will be the current risk-free rate plus the product of Happy Corp.'s beta and the market risk premium.
To plot the new Security Market Line (SML) on the graph, we would use the new required return calculated above and plot it against the corresponding beta values. The SML represents the relationship between risk (beta) and return (required rate of return).
By incorporating the new required return, we can determine the new expected returns for various levels of beta and create the updated SML.
It is important to note that without specific values provided for the risk-free rate, market risk premium, and Happy Corp.'s beta, it is not possible to calculate the exact new required return or plot the new SML accurately.
These values are crucial in determining the precise position of the SML on the graph.
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B
A
C
Intro
y
-6
4
3
2
+
1
2 3
x
Suppose quadrilateral ABCD has been transformed by
Ty=x. What are the coordinates for the vertices of the
reflected quadrilateral A'B'C'D'?
A' =
B' =
C' =
D'=
The coordinates of the reflected quadrilateral A'B'C'D' are:
A' = (6, 4)
B' = (-3, 2)
C' = (-1, 23)
D' = (-x, 12)
To find the coordinates of the reflected quadrilateral A'B'C'D', we need to apply the transformation Ty = x to each vertex of the original quadrilateral ABCD. The transformation Ty = x reflects each point across the y-axis.
Given the coordinates of the original quadrilateral ABCD as:
A = (-6, 4)
B = (3, 2)
C = (+1, 23)
D = (x, 12)
Applying the transformation Ty = x to each vertex, we can determine the coordinates of the reflected quadrilateral A'B'C'D':
A' = (-(-6), 4) = (6, 4)
B' = (-3, 2)
C' = (-1, 23)
D' = (-x, 12)
The reflected quadrilateral A'B'C'D' thus has the following coordinates:
A' = (6, 4)
B' = (-3, 2)
C' = (-1, 23)
D' = (-x, 12)
Therefore, the x-coordinate for point D' will be represented as -x in the reflected quadrilateral.
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AB=BC
A
60°
ODC
D
AB
374
B
The longest segment shown is
BC
C
Note that the longest segment in the shapes shown is DC (Option B).
How is this so?The longest side of a triangle is opposite to greatest angle.
To determine the longest side in a triangle, compare the lengths of all three sides. The side with the greatest length is the longest side.
You can use a ruler or a measuring tool to measure the lengths of the sides or compare the numerical values if they are provided.
In this case,
∠A = ∠DBA = 60°
So ∠ ABD is an equilateral triangle.
So, AB = BD = AD
Since
AB = BC
Then
∠BDC = ∠C ∠ 38°
so ∠DBC > 90°
This means that DC is the longest side.
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Solve for x leave your answer in simplest radical form
Answer:
X=11 trust me on my mom
Select the correct answer.
Which expression is equivalent to
OA. 5 (¹
OB.
5 (x¹ - 4x² + 3)
2¹-4²+3
O c. 24
O D. 2¹-2²+3
4x² + 3
1
+3²? Assume that the denominator does not equal zero.
Answer:
B
Step-by-step explanation:
[tex]\frac{x^6-4x^4+3x^2}{5x^2}[/tex]
factor out the common factor of x² from each term on the numerator
= [tex]\frac{x^2(x^4-4x^2+3)}{5x^2}[/tex] ( cancel x² on numerator/ denominator )
= [tex]\frac{x^4-4x^2+3}{5}[/tex]
Determine the percentile of 6.2 using the following data set.
4.2 4.6 5.1 6.2 6.3 6.6 6.7 6.8 7.1 7.2
Your answer should be an exact numerical value.
The percentile of 6.2 is
%.
The percentile of 6.2 in the given dataset is 30%. This means that 30% of the values in the dataset are lower than or equal to 6.2.
To determine the percentile of 6.2 in the given dataset, we need to calculate the percentage of values in the dataset that are lower than or equal to 6.2.
First, we arrange the dataset in ascending order: 4.2, 4.6, 5.1, 6.2, 6.3, 6.6, 6.7, 6.8, 7.1, 7.2.
Next, we count the number of values that are lower than or equal to 6.2. In this case, there are three values: 4.2, 4.6, and 5.1.
The next step is to calculate the percentage. We divide the count (3) by the total number of values in the dataset (10) and multiply by 100.
(3/10) * 100 = 0.3 * 100 = 30%
Percentiles are used to understand the relative position of a particular value within a dataset. In this case, 6.2 is higher than 30% of the values in the dataset and lower than the remaining 70%.
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What is the value of the expression (-2)(3)º(4)-2 ?
A. -3/2
B. -1/2
C. -3/4
D. 0
The value of the expression (-2)(3)º(4) - 2 is -164.
Based on the answer choices provided, none of the options matc.
To solve the expression (-2)(3)º(4)-2, we need to follow the order of operations, which is parentheses, exponents, multiplication, and subtraction.
Let's break down the expression :
(-2)(3)º(4) -2
First, we calculate the exponent:
(-2)(81) - 2
Next, we perform the multiplication:
-162 - 2
Finally, we subtract:
-164
Therefore, the value of the expression (-2)(3)º(4) - 2 is -164.
Based on the answer choices provided, none of the options match the value of -164.
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write 718000 in standard form
Answer:
718000
Step-by-step explanation:
718000 is already in standard form
Evaluate. -15 +7-(-8)
The answer options are
16
0
-16
-3
Answer:
To evaluate -15 + 7 - (-8), we can simplify the expression by first removing the double negative.
-15 + 7 + 8 = 0
Therefore, the answer is 0.
Step-by-step explanation:
The answer is:
0Work/explanation:
Remember the integer rule,
[tex]\bullet\phantom{4444}\sf{a-(-b)=a+b}[/tex]
Similarly
[tex]\sf{-15+7-(-8)}[/tex]
[tex]\sf{-15+15}[/tex]
Simplify fully.
[tex]\sf{0}[/tex]
Therefore, the answer is 0.Please answer ASAP I will brainlist
(a) The amount for total expenditures in 2015 was about $63.2 billion.
(b) The first full year in which expenditures exceeded $110 billion was 2027.
(a) To find the amount for total expenditures in 2015, we need to substitute x = 20 into the function h(x) = [tex]23.4(1.08)^{(x-5)[/tex].
h(x) = 23.4(1.0[tex]8)^{(20-5)[/tex]
= 23.4(1.0[tex]8)^{15[/tex]
≈ 23.4(2.717)
Rounding to the nearest tenth, the total expenditures in 2015 were about $63.2 billion.
(b) To determine the first full year in which expenditures exceeded $110 billion, we need to find the value of x when h(x) is greater than $110 billion.
110 = 23.4(1.0[tex]8)^{(x-5)[/tex]
Dividing both sides by 23.4:
4.7008547... = (1.0[tex]8)^{(x-5)[/tex]
Taking the logarithm base 1.08 of both sides:
log₁.₀₈(4.7008547...) = x - 5
Using a logarithm calculator or software, we find:
x - 5 ≈ 11.75
Adding 5 to both sides:
x ≈ 16.75
Rounding to the nearest whole number, the first full year in which expenditures exceeded $110 billion was 2027.
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how many pattern block rhombuses would 4 triangles create?
With 4 triangles, you can create a total of 3 pattern block rhombuses, depending on their arrangement.
To determine the number of pattern block rhombuses that can be created using 4 triangles, let's start by understanding the properties and arrangement of these shapes.
Pattern block rhombuses are a type of geometric shape commonly used in mathematics education. Each rhombus is made up of 2 triangles, specifically two congruent (equal) acute triangles. The triangles are placed together in a specific way to form the rhombus shape.
When 4 triangles are used, they can be arranged in different configurations to create different numbers of pattern block rhombuses. Let's explore the possibilities:
Arrangement 1:
In this arrangement, you can create 2 pattern block rhombuses. The triangles are placed side by side, with two triangles forming one rhombus, and the other two triangles forming another rhombus.
Arrangement 2:
In this arrangement, you can create 1 pattern block rhombus. The triangles are placed on top of each other, forming a larger triangle. Since a pattern block rhombus requires two acute triangles, only one rhombus can be formed in this case.
So, with 4 triangles, you can create a total of 3 pattern block rhombuses, depending on how the triangles are arranged.
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. Initially 100 milligrams of a radioactive substance was present.
After 6 hours the mass had decreased by 3%. If the rate of
decay is proportional to the amount of the substance present at
time t, nd the amount remaining after 24 hours.
Answer:
Incomplete Question
Solve the system of equations. 8 � + 5 � = 24 � = − 4 � 8x+5y=24 y=−4x
The solution to the system of equations is x = 2 and y = -8.
To solve the system of equations, we'll use the substitution method. The given equations are:
Equation 1: 8x + 5y = 24
Equation 2: y = -4x
We'll substitute Equation 2 into Equation 1 to eliminate one variable:
8x + 5(-4x) = 24
8x - 20x = 24 [Distribute the -4]
-12x = 24 [Combine like terms]
x = 24 / -12 [Divide both sides by -12]
x = -2
Now that we have the value of x, we can substitute it back into Equation 2 to find the value of y:
y = -4(-2)
y = 8
Therefore, the solution to the system of equations is x = -2 and y = 8.
However, let's double-check the solution by substituting these values into the original equations:
Equation 1: 8(-2) + 5(8) = 24
-16 + 40 = 24
24 = 24 [LHS = RHS, equation is satisfied]
Equation 2: 8 = -4(-2)
8 = 8 [LHS = RHS, equation is satisfied]
Both equations are satisfied, confirming that x = -2 and y = 8 is indeed the solution to the given system of equations.
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Roger can run one mile in 9 minutes. Jeff can run one mile in 6 minutes. If Jeff gives Roger a 1 minute head start, how
long will it take before Jeff catches up to Roger? How far will each have run?
Not including the head start, it will take
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minutes for Jeff to catch up to Roger.
Each person runs 1/3 of a mile when Jeff catches up to Roger.
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Explanation
x = number of minutes that Jeff runs
x+1 = number of minutes Roger runs
Roger has the head start of 1 minute, so he has been running for 1 minute longer compared to Jeff.
Roger runs 1 mile in 9 minutes. His unit rate is 1/9 of a mile per minute.
Jeff's unit rate is 1/6 of a mile per minute.
Let's set up a table with what we have so far
[tex]\begin{array}{|c|c|c|c|} \cline{1-4} & \text{Distance} & \text{rate} & \text{time}\\\cline{1-4}\text{Jeff} & d & 1/6 & \text{x}\\\cline{1-4}\text{Roger} & d & 1/9 & \text{x}+1\\\cline{1-4}\end{array}[/tex]
The distance equation for Jeff is d = (1/6)x
The distance equation for Roger is d = (1/9)(x+1)
note: distance = rate*time
Both runners travel the same distance when Jeff catches up to Roger, so both "d"s are the same value at this specific moment. Set the right hand sides equal to each other and solve for x.
(1/6)x = (1/9)(x+1)
18*(1/6)x = 18*(1/9)(x+1)
3x = 2(x+1)
3x = 2x+2
3x-2x = 2
x = 2
Jeff runs for 2 minutes when he catches up to Roger.
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Check:
Jeff runs for 2 minutes, at 1/6 of a mile per minute, so he runs 2*(1/6) = 2/6 = 1/3 of a mile.
Roger runs for 2+1 = 3 minutes (remember he gets the head start) at 1/9 of a mile per minute, so he has run 3*(1/9) = 3/9 = 1/3 of a mile as well.
Both men have run the same distance which confirms Jeff catches up to Roger at this point. The answer is confirmed.