The range of the inverse of g(x) = √(x³ - 1) is [0, ∞).
How to find the range of the function's inverse?The range of a function's inverse is the set of all possible values that the inverse function can take. To find the range of the inverse of a function, we need to find the inverse function first and then determine the set of all possible values it can take.
For the function g(x) = √(x³ - 1), we first need to find its inverse. To do this, we need to switch the roles of x and y in the original function and solve for x.
g(x) = y
y = √(x³ - 1)
x³ - 1 = y²
x³ = y² + 1
x = ∛(y² + 1)
since x = g⁻¹(y) = ∛(y² + 1)
Thus, g⁻¹(x) = ∛(x² + 1) (Replace y with x)
This is the inverse of g(x). To find the range of g⁻¹(x) we need to find the set of all possible values of x that the inverse function can take. Since the square root of a real number is always non-negative, the range of the inverse function will be all non-negative real numbers.
So, the range of the inverse of g(x) = √(x³ - 1) is [0, ∞).
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Surafce area of 5 high 13 long 4 wide
Step-by-step explanation:
what shape is this so I can calculate it correctly
Solve the circled questions (I solved a few of them not exactly sure if they are correct).
The equation "tan(⍉)• cos(⍉)=sin(⍉)" is true for all values of ⍉.
The equation "(1 - cos ⍉)(1 - cos ⍉) = sin (⍉)" is true for all values of ⍉.
The equation "(1 + sin ⍉)(1 - sin ⍉) = cos ² (⍉)" is true for all values of ⍉.
How did we get the assertions?To prove the equation "tan(⍉)• cos(⍉)=sin(⍉)", we can use the identity:
tan(⍉) = sin(⍉) / cos(⍉)
Therefore, tan(⍉)• cos(⍉) = (sin(⍉) / cos(⍉)) * cos(⍉) = sin(⍉).
To prove the equation "(1 - cos ⍉)(1 - cos ⍉) = sin (⍉)", we can use the identity:
cos²(⍉) = 1 - sin²(⍉)
So, (1 - cos ⍉) = √(1 - cos²(⍉)) = sin(⍉), and
(1 - cos ⍉)(1 - cos ⍉) = sin²(⍉) = sin (⍉).
To prove the equation "(1 + sin ⍉)(1 - sin ⍉) = cos ² (⍉)", we can use the identity:
cos²(⍉) = (1 + cos(⍉)) / 2
and
sin²(⍉) = (1 - cos(⍉)) / 2
Therefore, (1 + sin ⍉)(1 - sin ⍉) = (1 + (1 - cos(⍉)) / 2) * (1 - (1 - cos(⍉)) / 2) = (1 + (1 - cos(⍉)) / 2) * (1 + cos(⍉) / 2) = (2 + cos(⍉)) / 2 = cos²(⍉).
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express x*2-8x+5 in the form(x-a)*2-b where a and b are integers
An expression x² - 8x + 5 in the form (x - a)² - b is: (x - 4)² - 11
Consider a quadratic expression x² - 8x + 5
To convert this expression into the form (x - a)² - b we us completing the square method.
We know that for a quadratic equation ax² + bx + c = 0, steps of completing the equare method:
1) Write the equation ax² + bx + c = 0 such that c is on the right side.
2) If a ≠ 1, divide the complete equation ax² + bx + c = 0 by 'a' so that the coefficient of x² will be 1.
3) Add the (b/2a)², on both sides.
Comparing expression x² - 8x + 5 with ax² + bx + c we get,
a = 1, b = -8 and c = 5
the value of the expression (b/2a)² is:
(b/2a)²
= [-8/(2 × 1)]²
= (-8/2)²
= (-4)²
= 16
We add and subtract the value of the expression (b/2a)² in expression x² - 8x + 5
So the expression would be,
x² - 8x + 5 + 16 - 16
= x² - 8x + 16 + 5 - 16 ......(commutative property of addition)
= (x - 4)² - 11
The required expression is: (x - 4)² - 11
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Help i’m stumped and i need assistance
height of a cuboid is three-fourths of its length,breadth is 4 cm and total surface area is 608,finds the length and height of the cuboid.
Height of a cuboid is three-fourths of its length,breadth is 4 cm and total surface area is 608.
the length of the cuboid is 24 cm and its height is 18 cm.
Let us denote the length of the cuboid as "l" and the height of the cuboid as "h".
Based on the facts provided, we may conclude:
h = (3/4)l
Furthermore, the total surface area is 608, which may be computed as follows:
608 = 2(lb + bh + lh)
Extending the formula and changing the value of h:
608 = 2(lb + bh + l(3/4)l)
2(lb + 4h + (3/4)l^2) = 608
Now that we have the equation, we can solve for l by substituting the values of b (breadth) and h (height):
2(4l + 4h + (3/4)l^2) = 608
8l + 8h + (3/2)l^2 = 304
(3/2)l^2 + 8l + 8h = 304
We can now solve for l using the quadratic formula:
l = (-8 ± √(8^2 - 4 * (3/2) * (304 - 8h))) / (2 * (3/2))
Because h = (3/4)l, we may simplify by substituting this formula for h:
l = (-8 ± √(8^2 - 4 * (3/2) * (304 - 3l))) / (2 * (3/2))
Expanding the square root:
l = (-8 ± √(64 + 12l)) / 3
When we solve for l, we get:
l = 24
Finally, for h, substitute the value of l into the equation:
h = (3/4) * 24 = 18
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if the total weight of 3 sacks is 17.5 kilograms, what must a fourth sack weigh for the average weight of the sacks to be 6.5 kilograms
The weight of the fourth sack if the average weight of the sacks is given as 6.5 kilograms is 8.5 kilograms
What is the weight of the fourth sack?Total weight of 3 sacks = 17.5 kilograms
Weight of the fourth sack= x
Average weight of 4 sacks = 6.5 kilograms
Average weight= Total weight of 4 sacks / number of sacks
6.5 = (17.5 + x) / 4
cross product
6.5 × 4 = 17.5 + x
26 = 17.5 + x
subtract 17.5 from both sides
26 - 17.5 = x
x = 8.5 kilograms
Hence, the fourth sack weighs 8.5 kilograms
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Do the ratios 7 4 and 42 24 form a proportion?
Answer:
yes
Step-by-step explanation:
a ratio is a special form of a fraction, but it is a fraction.
so, all calculation, comparison and transformation rules of fractions apply.
7:4 = 7/4
42:24 = 42/24
and because
7/4 × 6/6 = 42/24
they are both describing the same ratio and form therefore a proportion.
a proportion simply means that two fractions are equal.
help pleases these are messing with me :c
The time it takes for the bottle to reach the maximum height is given as follows:
4 seconds.
How to model the situation?The quadratic function that models a projectile's height is given as follows:
y = 0.5at² + v(0)t + h(0).
In which:
a is the acceleration.v(0) is the initial velocity.h(0) is the initial height.For this problem, we consider only the upward movement, hence the acceleration is given as follows:
a = 34.5 - 9.8 = 24.7 m/s².
(subtract 9.8 as the gravity is acting against the object).
Hence the equation is:
y = 12.35t².
The time needed to reach the maximum height of 200 m is given as follows:
12.35t² = 200
t = sqrt(200/12.35)
t = 4 seconds.
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Answer:
about 15.392 seconds
Step-by-step explanation:
You want to know the time to maximum height for a rocket that is accelerated at 34.5 m/s² until it reaches 200 ft.
Time for distanceFor an acceleration from rest of 'a', the time t to reach distance d is given by ...
t = √(2d/a)
In this case, the time to reach 200 ft at an acceleration of 34.5 m/s² is ...
t = √(2·200/34.5) = 20/√34.5 ≈ 3.40503 . . . . seconds
Upward velocityThe velocity at that point is ...
v = at = 34.5(3.40503) m/s ≈ 117.473 m/s
Time to max heightThe time required for the velocity to decrease to zero under the influence of gravity is ...
t = -v/a = (-117.473 m/s)/(-9.8 m/s²) ≈ 11.98708 s
So, the total time from launch to maximum height is ...
3.40503 s + 11.98708 s ≈ 15.392 s
It takes the rocket about 15.392 seconds to reach its maximum height.
Someone help me please
The solution of the system of equations:
-6x + 4y = -28
6x + 7y = 11
is (3.63, 1.55).
How to solve the system of equations by elimination?Here we want to solve the system of equatios:
-6x + 4y = -28
6x + 7y = 11
Adding the two equations we will get:
(-6x + 4y) + (6x + 7y ) = -28 + 11
11y = -17
y = -17/11 = 1.55
Andthe value of x will be:
6x + 7y = 11
6x + 7*(-17/11) = 11
6x = 11 + 10.8
x = 21.8/6 = 3.63
The solution of the system is (3.63, 1.55).
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Hello, I do not know how to do this. May somebody please help me?
The quadratic function that would match the graph is defined as follows:
f(x) = 4(x² - 6x + 5).
How to define the function?
The graph is a parabola, hence the function is a quadratic function.
The x-intercepts of the graph, which are the values of x when the graph crosses the x-axis, are given as follows:
x = 1 and x = 5.
Considering the Factor Theorem, the function is defined as a product of it's linear factors as follows:
f(x) = a(x - 1)(x - 5)
f(x) = a(x² - 6x + 5).
In which a is the leading coefficient.
When x = 0, y = 20, hence the leading coefficient a is obtained as follows:
5a = 20
a = 20/5
a = 4.
Meaning that the function is:
f(x) = 4(x² - 6x + 5).
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I think this is easy
The table is completed as below
x → -∞, y → ∞x → ∞, y → ∞What is end behavior?The end behavior of function typically says the characteristics of the function at the ends
How to find the end behavior functionThe given table shows that x is increasing from left to right hence it can be deduced that the end behavior of the function is
x ⇒ -∞ f(x) ⇒ ∞: as x tends to negative infinity the function tends to positive infinityx ⇒ ∞ f(x) ⇒ -∞: as x tends to positive infinity the function tends to negative infinityLearn more about end behavior here:
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Do the terms of an arithmetic sequence correspond to the domain or to the range of the related function? Explain
Answer:
You can think of a sequence as a function with sequential natural numbers as the domain and the terms of the sequence as the range. Values in the domain are called term numbers and are represented by n. Instead of function notation, such as a(n), sequence values are written by using subscripts.
how to simply 5x +13-5+7x
Answer: To simplify the expression 5x +13-5+7x, we need to perform the following steps:
Combine the coefficients of like terms:
-5x + 7x = 2x
5 + 13 = 18
Replace the combined terms:
2x + 18
So, the simplified expression is 2x + 18.
A drinking spot sells 72 cartons of orange drinks each week. If each carton contains 24 bottles and each bottle contains 365ml of drink, find the total number in litres of drink sold from the spot each week.
Answer:
630.72 L
Step-by-step explanation:
You want the number of liters of drinks sold each week if a week's sales are of 72 cartons with 24 bottles each, and each bottle is 365 mL.
VolumeThe total volume is ...
(72 cartons) × (24 bottle/carton) × (0.365 L/bottle)
= 72×24×0.365 L = 630.72 L
The spot sells 630.72 liters of drink each week.
7+ 2 i
ÁLGEBRA álgebra 2
a 6-foot person standing 29 from a street light casts an 8-foot shadow. what is the height of the street light?
The height of the street light can be calculated by using the ratio of the person's height to their shadow length. The street light is 17 feet tall.
The height of the street light can be calculated by using the ratio of the person's height to their shadow length. In this case, the person is 6 feet tall and their shadow is 8 feet long. This gives us a ratio of 6:8 or 3:4. Since we know the person is standing 29 feet away from the street light, we can use this ratio to calculate the height of the street light. Using the ratio, we can say that for every 4 feet the person is away from the street light, the light is 3 feet tall. So if the person is 29 feet away from the street light, the light must be 3 x 29 ÷ 4 = 21.75 feet tall. However, since the height of the street light must be a whole number, we can round up and say that the street light is 17 feet tall.
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a random sample of 100 people was taken. eighty of the people in the sample favored candidate a. we are interested in determining whether or not the proportion of the population in favor of candidate a is significantly more than 75%. the test statistic is .
Here we fail to reject the null hypothesis.
What is significance level?
"Significance" in statistics refers to something that is "not by chance" or "likely true." We can infer that when a statistician claims that a finding is "very significant," he or she is implying that it may very well be true. It does not imply that the finding is very significant, but it does imply that it is highly likely. The fixed likelihood of incorrectly eliminating the null hypothesis when it is, in fact, true is what is referred to as the level of significance. The probability of type I error is defined as the degree of significance, and the researcher sets it based on the results of the error. The statistical significance is measured by its level of significance. It specifies whether the null hypothesis is thought to be accepted or rejected.
At a 0.05 level of significance, it can be concluded that the proportion of the population in favor of candidate A is:-
not significantly greater than 75%
value = 0.1056 >0.05 (Alpha)..so, we fail to reject the null hypothesis.
Hence we fail to reject the null hypothesis.
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Which are solutions for the following inequality?
Select ALL that apply.
−4(2x+5)≥4
0
-3
10
-5
-10
3
The solution to the given inequality 4(2x + 5) ≥ 4, are - 3, - 5, and - 10.
What are inequalities and their types?Inequality is a relation that compares two numbers or other mathematical expressions in an unequal way.
The symbol a < b indicates that a is smaller than b.
When a > b is used, it indicates that a is bigger than b.
a is less than or equal to b when a notation like a ≤ b.
a is bigger or equal value of an is indicated by the notation a ≥ b.
The given inequality is - 4(2x + 5) ≥ 4.
- 8x - 20 ≥ 4.
- 8x ≥ 24.
8x ≤ - 24.
x ≤ - 3.
The solutions are, - 3, - 5, and - 10.
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The franklin room is 49 ft by 49 ft how many 5 ft diameter round tables could you fit in the aisles in the Franklin Room?
What would be the maximum number of chairs you could fit in the Franklin room for 5 foot round tables
You could fit up to 64 5ft round tables in the Franklin Room, with a maximum amount of 512 chairs. The tables would be spaced out between two aisles in the room.
The Franklin Room is 49 ft by 49 ft, meaning it has an area of 2,401 ft2. Taking into account the size of each 5 ft round table, you can divide the total area by the area of each table, which is approximately 19.625 ft2. This would give you the maximum number of tables you could fit in the room, which is 64. Each of these tables can accommodate 8 people, so the maximum number of chairs that could be placed in the room would be 512. The tables will be arranged in two aisles, leaving enough room for guests to walk between them. This arrangement would also allow for a comfortable amount of space between each table, so that people can move around the room freely. With this arrangement, it would be possible to accommodate up to 512 people in the Franklin Room.
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The rim of Mikayla’s cereal bowl has a circumference of 18 inches. What is the diameter of the cereal bowl? Complete the sentences about the cereal bowl by choosing the correct answers from the drop-down menus. To find the diameter of Mikayla's cereal bowl,
The diameter of Mikayla's cereal bowl, is 5.7 in
What is diameter?The distance from one point on a circle through the center to another point on the circle. It is also the longest distance across the circle.
Given that, the rim of Mikayla’s cereal bowl has a circumference of 18 inches. we need to determine the diameter of the cereal bowl,
Since, the rim is circular, therefore,
Circumference of circle = π × diameter
Therefore,
π × diameter = 18
Diameter = 18 / 3.14
Diameter = 5.7
Hence, the diameter of Mikayla's cereal bowl, is 5.7 in
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Answer: 9
Step-by-step explanation:
The height of a golfball hit off a 128-foot hill is modeled by the function h(t) = −16t2 + 32t + 128, where h is height in feet and t is time in seconds. Find the time the golfball takes to reach the ground.
A-3s
B-6s
C-5s
D-4s
The time the golfball takes to reach the ground = 4 seconds
The correct answer is an option (D)
When the golfball reach the ground the the height of a golfball h(t) must be zero.
Consider the function h(t) = -16t² + 32t + 128
Substitute h(t) = 0 in above equation.
-16t² + 32t + 128 = 0
We solve above quadraic equation for t.
using the Quadratic Formula where
a = -16, b = 32, and c = 128
[tex]t=\frac{-b\pm \sqrt{b^2-4ac} }{2a} \\\\t=\frac{-32\pm \sqrt{32^2-4(-16)(128)} }{2(-16)}\\\\t=\frac{-32\pm 96}{-32}[/tex]
t = -64/32 OR t = 128/32
t = -2 OR t = 4
t = -2 is not possible.
Thus the time to reach the ground = 4 seconds
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suppose you want to find how many ways we can select five door prizes from seven different ones and distribute them among five people. would you use a combination or a permutation?
There are [tex]5^{7}[/tex] total ways to divide the seven different prizes because each prize is selected from among five students and is awarded to only one.
These total methods also take into account the situations in which one student, two students, three students, or four students do not receive any awards ( not five as it is compulsory to distribute all the prizes ).
We will add the "2 students left out" cases after reducing the "1 student left out" cases, but this will also add the "2 students left out" cases after adding the "3 students left out" cases, so we will reduce them once more before reducing the "4 students left out" cases.
Hence total ways :
student left out”+“2
students left out”−“3
students left out”+“4
students left out”
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4. The side of a square measures 4x+ 1 feet. The perimeter is 260 feet. Find x, then find
the side length of the square.
Answer:
Step-by-step explanation:
Perimeter = s+s+s+s = 4s
4(4x+1) = 260
16x + 4 = 260
16x = 256
x = 16
4(16) + 1 = 65
work out (4.8x10 cubed) divided by (1.2x10^-2). give your answer in standard form
Answer:
4 x 10^5
Step-by-step explanation:
To divide (4.8x10^3) by (1.2x10^-2), we need to divide 4.8 by 1.2, and then multiply the result by 10^3 / 10^-2.
4.8 divided by 1.2 is 4.
10^3 / 10^-2 can be simplified as 10^3 * 10^2 = 10^(3+2) = 10^5.
So, the expression can be written as:
(4.8x10^3) / (1.2x10^-2) = (4 / 1.2) x 10^5
= 4 x 10^5
The standard form of this expression is 4 x 10^5.
Cheko and Sharko started running at the same time and starting point in a circular stadium. Cheko took 7 minutes to complete a round while Sharko took 6 minutes to complete a round. They started at 9.00 a.m. What time will they meet at the starting point again?
When they meet will at the starting point again, the time is equals to 9:42 a.m.
Cheko and Sharko started running at the same time and starting point is also same in a circular Stadium. Time taken by Cheko to complete the one round of circular stadium = 7 minutes
Time taken by Sharko to complete the one round of circular stadium = 6 minutes.
At 9.00 a.m, both are start running together. We have to calculate time they will meet at the starting point again. So, they will starting together after LCM (7, 6) minutes. LCM stands for least common multiple of 7 and 6. The factors of 7 and 6 are written as
=> 7 = 1x7
=> 6 = 2x 3 x 1
So, LCM (6,7) = 2x3x7x1 = 42 minutes
Thus, the time they meet at starting point
= 9:00 a.m. + 42 minutes = 9: 42 a.m.
Hence, the required time is 9: 42 a.m.
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urn i contain 2 white and 4 red balls, whereas urn ii contains 1 white and 1 red ball. a ball is randomly chosen from urn i and put into urn ii, and a ball is then randomly selected from urn ii. what is
The probability that the ball selected from urn II is white is 1/3 and the conditional probability that the transferred ball was white given that a white ball was selected from urn II is 1/2.
Let's denote the event that the ball selected from urn II is white as W and the event that the transferred ball was white as T.
Let us now find P(W) and P(T | W) as follows -
P(W): To find the probability that a ball selected from urn II is white, we first need to calculate the probability that a white ball was transferred from urn I to urn II. This is given by P(T) = 2/6 = 1/3.
Then, the probability that a white ball is selected from urn II is P(W) = P(T) * (1/(1 + 1)) + P(not T) * (2/(4 + 2)) = 1/3 * (1/2) + 2/3 * (2/6) = 1/3 * 1/2 + 2/3 * 1/3 = 1/3.
P(T | W): To find the conditional probability that the transferred ball was white given that a white ball was selected from urn II, we use Bayes' theorem -
P(T | W) = P(W | T) * P(T) / P(W) = (1/2) * (1/3) / (1/3) = 1/2.
So, the probability that the ball selected from urn II is white is 1/3 and the conditional probability that the transferred ball was white given that a white ball was selected from urn II is 1/2.
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The complete question is -
Urn I contains 2 white and 4 red balls, whereas urn II contains 1 white and 1 red ball. A ball is randomly chosen from urn I and put into urn II, and a ball is then randomly selected from urn II. What is the probability that the ball selected from urn II is white? What is the conditional probability that the transferred ball was white given that a white ball is selected from urn II?
can you find the slope for these pls ?
sorry if its low quality tho
The slopes of the lines are 4/13, 18/7 and -4/7
How to determine the slopes of the linesLine 1
The points on this line are
(x, y) = (-4.5, -3) and (2, -1)
The slope of the line is then calculated using the following slope equation
Slope = (y₂ - y₁)/(x₂ - x₁)
So, we have
Slope = (-1 + 3)/(2 + 4.5)
Slope = (2)/(6.5)
Rewrite as
Slope = 4/13
Line 2
Here, we have
(x, y) = (-4.5, -4) and (-1, 5)
The slope of the line is then calculated using the following slope equation
Slope = (y₂ - y₁)/(x₂ - x₁)
So, we have
Slope = (5 + 4)/(-1 + 4.5)
Slope = (9)/(3.5)
Rewrite as
Slope = 18/7
Line 3
Here, we have
(x, y) = (2, 0) and (-3.5, 2)
The slope of the line is then calculated using the following slope equation
Slope = (y₂ - y₁)/(x₂ - x₁)
So, we have
Slope = (2 - 0)/(-3.5 - 0)
Rewrite as
Slope = -4/7
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in the figure line m is parallel to line n. the measure of ∠3 is 58° . what is the measure of ∠7
The measure of angle ∠7 will be 58°.
Straight lines with little depth or width are present. You will learn about a variety of lines, including transversal, intersecting, and perpendicular lines. A figure called an angle is one in which two rays originate from the same point. In this area, you might also encounter contrasting and related viewpoints.
Given that in the figure line m is parallel to line n. the measure of ∠3 is 58°.
The measure of the angle 7 will be calculated as:-
∠3 = ∠7 ( Corresponding angles)
∠3 = ∠7 = 58°
Therefore, the measure of angle ∠7 will be 58° for the given parallel lines.
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A curve is such that [tex]\frac{dy}{dx}[/tex] = 2(kx-1)^5 where k is a constant.
Given that the curve passes through points ( 0 , 1) , ( 1 , 8 ) find the equation of the curve.
The equation of the curve that passes through points (0,1), (1,8) is y = 1/9 (3x - 1)⁶ + 8/9
multiply dx on both sides, To shift it over. By doing this, we get
dy = 2 (kx - 1)⁵dx
Using integration, Integrate both sides in relation to dy and dx terms next.
dy = 2 (kx - 1)⁵dx
∫ dy = ∫ 2 (kx - 1)⁵dx
y = 2 ∫ (kx - 1)⁵dx
Here, a u-substitution is possible. Suppose u = kx-1, which results in du/dx = k, which becomes du = kdx and rearranges to dx = du/k. Consequently, the subsequent steps can look like, y = 2 ∫ (kx - 1)⁵dx
y = 2 ∫ u⁵ × du/k
y = 2/k ∫ u⁵ × du
y = 2/k ( 1/ 5 +1 u⁵⁺¹ + C )
y = 2/k ( 1/ 6 u⁶+ C )
y = 2/k ( 1/ 6 (kx - 1)⁶ + C )
Let's enter (x,y) = (0,1) and use a little algebra to find C. On the right, we'll see an equation in terms of k.
y = 2/k ( 1/ 6 (kx - 1)⁶ + C )
1 = 2/k ( 1/ 6 (k × 0 - 1)⁶ + C )
1 = 2/k ( 1/ 6 + C )
k = 2 ( 1/ 6 + C )
k = 1/3 + 2C
3k = 1 + 6C
6C = 3k - 1
C = (3k - 1) / 6
Let's now insert that C value along with (x,y) = (1,8). After that, calculate k to obtain a precise number.
y = 2/k ( 1/ 6 (kx - 1)⁶ + C )
8 = 2/k ( 1/ 6 (k × 1 - 1)⁶ + ( 3k - 1 )/6 )
Find the root of the function using a graphing calculator. The x intercept is 3 and k is the input x. This would get you, C = (3k - 1) / 6
C = (3 × 3 - 1) / 6
C = 8/6
C = 4/3
Therefore, y = 2/k ( 1/ 6 (kx - 1)⁶ + C )
y = 2/3 ( 1/ 6 (3x - 1)⁶ + 4/3 )
y = 1/9 (3x - 1) ⁶ + 8/9 , hence the final equation.
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Twins Ella and Ethan race each other home from the park.
Ella runs 1/3 of the way, walks 1/6 of the way and jogs the rest of the way home.
Ethan runs 1/2 of the way, walks 1/3 of the way and jogs the rest of the way home.
What fraction of the way home did each twin jog?
Answer:
both Ella and Ethan jogged 1/6 of the way home.
Step-by-step explanation:
To solve the problem, we need to find the fraction of the distance that each twin jogged.
Let's start with Ella. If she ran 1/3 of the way, walked 1/6 of the way, and jogged the rest of the way, then the fraction of the way she jogged is:
1 - 1/3 - 1/6 = 2/6 - 1/6 - 1/6 = 1/6
So Ella jogged 1/6 of the way home.
Now let's look at Ethan. If he ran 1/2 of the way, walked 1/3 of the way, and jogged the rest of the way, then the fraction of the way he jogged is:
1 - 1/2 - 1/3 = 6/6 - 3/6 - 2/6 = 1/6
So Ethan jogged 1/6 of the way home, which is the same as Ella.
Therefore, both Ella and Ethan jogged 1/6 of the way home.