A boat traveled 252 miles downstream and back. The trip downstream took 12 hours. The trip back took 84 hours. What is the speed of the boat in still water? What is the speed of the current?

The inverse function of f(x) = 6x^4 is f^(-1)(x) = (x/6)^(1/4), and f^(-1)(x) is a function.

To find the inverse function of f(x) = 6x^4, we can follow these steps:

1. Replace f(x) with y:
y = 6x^4

2. Swap x and y:
x = 6y^4

3. Solve for y:
Divide both sides by 6:
x/6 = y^4

4. Take the fourth root of both sides:
(x/6)^(1/4) = y

Now we have found the inverse function:
f^(-1)(x) = (x/6)^(1/4)

To determine if f^(-1)(x) is a function, we need to check if it passes the vertical line test. The vertical line test states that if a vertical line intersects a graph in more than one point, then the graph is not a function.

Let's imagine graphing f^(-1)(x) = (x/6)^(1/4). We see that for every x-value, there is only one corresponding y-value. Hence, each vertical line will intersect the graph at most once, confirming that f^(-1)(x) is indeed a function.

In conclusion, the inverse function of f(x) = 6x^4 is f^(-1)(x) = (x/6)^(1/4), and f^(-1)(x) is a function.

Final answer:

To solve for the number of dimes and quarters Michael has, we use two equations: 0.10d + 0.25q = 4.40 for the total value, and d + q = 20 for the total quantity of coins.

Explanation:

To find the number of dimes, d, and the number of quarters, q, Michael has, we need two equations to form a system. The first equation will represent the total amount of money in terms of dimes and quarters, and the second one will represent the total number of coins.

The value of a dime is $0.10 and the value of a quarter is $0.25. The total amount of money, $4.40, can be represented as 0.10d + 0.25q = 4.40. This is the value equation.

Since there are 20 coins in total, the second equation will simply be the count of both types of coins together: d + q = 20. This is the quantity equation.

The system of equations to find the number of dimes and quarters is:

0.10d + 0.25q = 4.40 (value equation)

d + q = 20 (quantity equation)

∠1 and ∠2 are right angles

What are Congruent Triangles?

Transformations change the size or position of shapes. Congruent shapes are identical, but may be reflected, rotated or translated. Scale factors can increase or decrease the size of a shape. Congruent Triangles simply mean the triangles that possess the same size and shape

•  The three sides are equal (SSS: side, side, side)

•  Two angles are the same and a corresponding side is the same (ASA: angle, side, angle)

•  Two sides are equal and the angle between the two sides is equal (SAS: side, angle, side)

•  A right angle, the hypotenuse and a corresponding side are equal (RHS, right angle, hypotenuse, side)

Given data ,

∠1 and ∠2 are supplementary

So , ∠1  + ∠2 = 180°

∠1 ≅ ∠2

So , ∠1 and ∠2 are congruent

Since , we know that both of these angles are congruent and supplementary , both of the angles add up to 180°

So , each of the angle measures a total of ∠1 = 180° / 2

                                                                           = 90°

And ∠2 = 90°

When the angle of a triangle is 90° , the angle is known as right angle

So ∠1  and ∠2 are right angles

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Final answer:

To calculate the fraction of a 10-hour shift that 6 hours represents, you divide 6 by 10 and simplify the resulting fraction, which is 6/10, to 3/5. Therefore, 6 hours is 3/5 of the work shift.

Explanation:

If an employee works a 10-hour shift, and you want to find out what fraction of the shift is represented by 6 hours, you divide 6 by the total hours in the shift which is 10. So, we write this as the fraction 6/10.

Now, simplifying the fraction, both the numerator (6) and the denominator (10) can be divided by 2. This gives us 3/5. Therefore, 6 hours represent 3/5 of the employee's 10-hour work shift.

To relate this to a concept you might already be familiar with, consider that 60 minutes make up 1 hour. If we write this as a ratio, it would be 60 minutes / 60 minutes, which simplifies to 1. Similarly, when we talk about fractions of an hour, like in the question, we're essentially looking for the number of parts of the shift that are being taken up by 6 hours in a 10-hour slot, which is 3/5.

The two numbers that represent the lengths of the legs of the right triangle are 2 and 5.

Trigonometry is the study of the relationship between the ratios of the sides of a right-angled triangle and their angles. The trigonometric ratios used to examine this relationship are sine, cosine, tangent, cotangent, secant, and cosecant.

Given:

A right triangle has an acute angle measure of 22°.

Using Trigonometry

tan [tex]\theta[/tex] = opposite/adjacent

tan 22° = 2/5

            = 0.40

Thus, the lengths of the legs of the right triangle are 2 and 5.

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Answer:

no its not

Step-by-step explanation:

simplest form for 22/33 is 2/3

Kelvin found the correct equation and solution.

Brittany write incorrect equation because difference of a number and 20 can be written as, x - 20 where, x is a number; but Brittany can be written as 20 - x.

Here,

Kevin and Brittany write an equation to represent the following relationship, and both students solve their equation.

And, condition is given as,

5 times the difference of a number and 20 is the same as half the sum of 4 more than 4 times a number.

What is linear equation?

A linear equation only has one or two variables. No variable in a linear equation is raised to a power greater than 1.

Now, according to given condition,

5 times the difference of a number and 20 is written as,

[tex]5 ( x - 20 )[/tex]

And, half the sum of 4 more than 4 times a number, is written as

[tex]\frac{1}{2} ( 4x + 4)[/tex]

Now, both equation are equal

[tex]5 ( x - 20 ) = \frac{1}{2} ( 4x + 4)\\5x - 100 = 2x + 2\\3x = 102\\x = 34[/tex]

Hence, Kelvin found the correct equation and solution.

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True.

4x-3=19

4x=21 (add 3)

x=5.25 (divide both sides by 4)

Answer:

Open

Step-by-step explanation:

i just took the quick check

trust me

To find out how much shorter Noam's walk would be if there were a direct path from the school to his house, we need to calculate the displacement using the Pythagorean theorem. Noam walks 8 blocks north and then 6 blocks east. The displacement is the straight-line distance from the starting point to the endpoint. Therefore, the displacement is the square root of (8² + 6²), which is the square root of 64 + 36, which is the square root of 100, which is 10 blocks. So, if there was a direct path, Noam's walk would be 10 blocks shorter.

To find out how much shorter Noam's walk would be if there were a direct path from the school to his house, we need to calculate the displacement using the Pythagorean theorem. Noam walks 8 blocks north and then 6 blocks east. The displacement is the straight-line distance from the starting point to the endpoint.

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Therefore, the displacement is the square root of (8² + 6²), which is the square root of 64 + 36, which is the square root of 100, which is 10 blocks. So, if there was a direct path, Noam's walk would be 10 blocks shorter.

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Answer: 80x130

Step-by-step explanation:

The percentage of red marbles in a bag is 65%.

Given that, a bag of marbles contains 7 blue marbles and 13 red marbles.

Percentage is defined as a given part or amount in every hundred. It is a fraction with 100 as the denominator and is represented by the symbol "%".

Here, total number of marbles in a bag

= 7+13 =20

Now, percentage of red marbles

= 13/20 × 100

= 65%

Therefore, the percentage of red marbles in a bag is 65%.

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Answer:   n6 + 7mn5 + 14m2n4 - 5m3n3 - 6m6

Step-by-step explanation:

divide 36 by 8

36 / 8 = 4.5

 so he had 4 pieces that were 8 feet long and he had 1 piece 4 feet long

 so he had 5 pieces of wire

The question is missing critical information as it lacks the proportion or number related to the shaded part of the model. Without this detail or a visual representation, it's not possible to determine the accuracy of Gary or John's statements.

The question provided seems to be missing critical information, specifically the proportion or number that Gary and John are referring to when they mention how much of a model is shaded. In a typical mathematics problem regarding fractions or percentages, students would be asked to determine if statements about the shaded portion of a model are correct. Without the visual model or additional context indicating the proportion of the model that is shaded, it's impossible to conclude which student, Gary or John, is correct.

In order to answer correctly, we would need either a clear description of the model and its shaded parts or, preferably, a visual representation. Then, we would analyze the shaded portion to assess which statement about the fraction or percentage shaded is accurate. If we had this information, we could illustrate the process of comparing the given fraction or percentage with the actual shaded section of the model.

Unfortunately, without the necessary details, we can neither confirm if Gary, John, or both are correct, nor can we properly address the question.

Answer:

1

Step-by-step explanation:

[tex]27 \cdot ((3^3)^{-1})[/tex]

We need to evaluate the given expression

3^3 is 3 times 3 times 3= 27

[tex]27 \cdot ((3^3)^{-1})[/tex]

[tex]27 \cdot ((27)^{-1})[/tex]

Now we apply exponential property

[tex]a^{-m}= \frac{1}{a^m}[/tex]

[tex]27^{-1}= \frac{1}{27}[/tex]

[tex]27 \cdot ((27)^{-1})[/tex]

[tex]27 \cdot \frac{1}{27}=1[/tex]