Name the quadrant in which sinθ < 0 and cosθ > 0.

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The value of the solution of expression is, r = 7

We have to give that,

An expression to simplify,

r + 15 = 4r - 6

Now, Simplify the expression by combining like terms as,

r + 15 = 4r - 6

Subtract r on both sides,

r + 15 - r = 4r - r - 6

15 = 3r - 6

Add 6 into both sides,

15 + 6 = 3r

21 = 3r

Divide 3 on both sides,

r = 21/3

r = 7

Therefore, the solution is, r = 7

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Answer:

Inverse function, [tex]f^{-1}=2^x[/tex]

Step-by-step explanation:

We are given a function [tex]f(x)=\log_2x[/tex]

We need to find the inverse of f(x)

Step 1: Set f(x)=y

[tex]y=\log_2x[/tex]

Step 2: Switch x and y

[tex]x=\log_2y[/tex]

Step 3: Solve for y (isolate y)

[tex]y=2^x[/tex]              [tex]\because \log_ab=x\Rightarrow b=a^x[/tex]

Inverse of function [tex]f(x)\rightarrow f^{-1}(x)=2^x[/tex]

The inverse of the logarithmic function is [tex]f^{-1}(x) = 2^x[/tex]

The logarithmic expression is given as:

[tex]f(x) = \log_2(x)[/tex]

Replace f(x) with y

[tex]y = \log_2(x)[/tex]

Swap the positions of x and y

[tex]x = \log_2(y)[/tex]

Apply the change of base rule of logarithm

[tex]2^x = y[/tex]

Rewrite as:

[tex]y = 2^x[/tex]

Express y as an inverse function

[tex]f^{-1}(x) = 2^x[/tex]

Hence, the inverse of the logarithmic function is [tex]f^{-1}(x) = 2^x[/tex]

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Let us say that:

x = the time it take for the new pump to drain the pool alone

4x = the time it take for the older pump

And we are given that together they drain the pool in 4 hours, therefore:

1/x + 1/4x = ¼

Multiply everything by x:

1 + ¼ = ¼ x

1.25 = 0.25 x

x = 5 hours

So, 4x = 20 hours

Hence it take the older pump 20 hours draining alone.

Final answer:

The older pump would take 3.2 hours, or 3 hours and 12 minutes, to drain the pool on its own, since it takes 4 times as long as the newer pump which takes 48 minutes alone.

Explanation:

If an older pump takes 4 times as long to drain a pool as a newer pump, and together they take 4 hours to drain the pool, we can find out how long it would take for the older pump to drain the pool on its own by setting up an equation based on their rates of work.

Let's denote the time it takes for the newer pump to drain the pool alone as t hours. This means that the older pump would take 4t hours to do the job on its own.

Working together for one hour, they would collectively drain 1/t + 1/4t of the pool. Since they can drain the entire pool in 4 hours working together, we can write the equation:

1/t + 1/4t = 1/4

Now, solving for t:

4 + 1 = t

4t + t = 4t

5t = 4t

5t = 4

t = 4/5

So, the newer pump takes 4/5 of an hour (which is 48 minutes) to drain the pool alone. Then the older pump, taking 4 times as long, would take 4 * 4/5, which is 16/5 or 3.2 hours (which is 3 hours and 12 minutes) to drain the pool on its own.

The width of the given rectangle is 33  inches.

A rectangle is a geometrical figure in which opposite sides are equal.

The angle between any two consecutive sides will be 90 degrees.

Area of rectangle = length × width.

Perimeter of rectangle = 2( length + width).

Let's say the width of the rectangle is W while the length is L.

Given that,

A rectangle has a width that is 3 more than twice the length.

So,

W = 2L + 3

And,

Perimeter = 2(L + W) = 96

L + W = 48

By substitution,

L + 2L + 3 = 48

3L = 45

L = 15

So,

W = 2(15) + 3 = 33

Hence "The width of the given rectangle is 33  inches".

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Answer:

160^8

Step-by-step explanation:

The base 8 representation of 112 is 160.

The base 8 representation of 112 is 160.

To convert a number to base 8, you can divide the number by 8 repeatedly and write down the remainders in reverse order.

Here's the step-by-step process:

Write down the remainders in reverse order: 160.

The number of different coloured crayons is 76.

Given that, Henry wants to see how many different coloured crayons are in the crayons box. There are 4 rows of 19 crayons.

We need to find how many different coloured crayons are there, assuming no duplicate colours.

In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.

Let the number of different coloured crayons be x.

Now, x=4×19

⇒x=76

Therefore, the number of different coloured crayons is 76.

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Answer:

11 bags

Step-by-step explanation:

Let x bags of fertilizer should be purchased.

The dimension of rectangular lawn is 410 feet by 65 feet.

The area of rectangle is given by

Area = length × width

Thus, the area of rectangular lawn is

Area = 410 × 65

Area = 26650

Now, one bag of fertilizer covers 2500 square feet of lawn.

Thus, we have

[tex]\frac{1}{2500}=\frac{x}{26650}\\\\2500x=26650\\\\\x=\frac{26650}{2500}\\\\x=10.66[/tex]

Hence, the required number of bags = 11.

divide cost by number of bottles

7/28 = 0.25 per bottle

check 28 * 0.25 = 7.00 

Answer:

2.99x + 12.99y = 43.92

Step-by-step explanation:

Here, x represents the number of pairs of socks purchased, and let y represents the number of blouses purchased.

Since, the cost of each pair of socks is $ 2.99 and the cost of a blouse is $ 12.99,

Thus, the total cost of x pairs of socks and y blouses = ( 2.99x + 12.99y ) dollars,

According to the question,

Total cost is $43.92,

⇒ 2.99x + 12.99y = 43.92

Which is the required equation.

To determine the value(s) of h for a consistent linear system, the augmented matrix must not have any row that implies a contradiction, like 0x + 0y = c where c is non-zero. The exact values of h depend on the matrix itself and usually involve row reduction to identify non-contradictory solutions. The provided examples do not appear to relate directly to a linear system matrix.

To determine the value(s) of h that result in the matrix representing a consistent linear system, we must ensure that there are no contradictions within the augmented matrix. If the matrix represents a system of linear equations, consistency is achieved when the system has at least one solution. A contradiction would occur if a row in the augmented matrix had all zeros in the coefficient part and a non-zero value in the last column, corresponding to an equation like 0x + 0y = c (where c is non-zero), which has no solution.

The exact values of h would be determined by the context of the specific matrix and its elements, usually involving row reduction to echelon form to identify possible contradictions. In some cases, specific values of h can lead to a row of all zeros, which does not affect consistency, but a particular non-zero value in the rightmost column would create inconsistency. Without the explicit matrix, we cannot provide the exact values of h.

In the examples provided, it seems there is a mix of mathematical equations and expressions that are not directly related to a specific matrix or linear system. As such, the value of h in those expressions may refer to concepts like the uncertainty principle in quantum mechanics or gravitational force in physics, and thus do not pertain to solving a system of linear equations.

Final answer:

After 30 seconds, she will be 150 meters away.

Explanation:

To calculate how far the friend will be after 30 seconds on her bike, we must assume a constant velocity since no acceleration is mentioned. Given that she is 15 meters away after 3 seconds, we can calculate her velocity using the formula v = d/t, where v is velocity, d is distance, and t is time.

First, we find her velocity:

v = 15 m / 3 s = 5 m/s

Now that we have her velocity, we can calculate how far she will be after 30 seconds:

Distance = Velocity × Time = 5 m/s × 30 s = 150 m

Therefore, after 30 seconds, she will be 150 meters away.

Thanks for the explanation mate that was helpful^^

The diving and swimming lessons will occur on the same day after 15 days because 15 is the smallest common multiple of 3 and 5.

The question pertains to finding a common multiple of the numbers 3 and 5. These numbers represent the intervals at which you have swimming and diving lessons respectively. The first common multiple of 3 and 5, other than 1, is 15. Therefore, you will have both lessons on the same day again after 15 days.

This comes from the idea that a number's multiples are the set of numbers formed by multiplying that number by any of the whole numbers. So, the multiples of 3 are 3, 6, 9, 12, 15, 18, etc., while the multiples of 5 are 5, 10, 15, 20, etc. The number 15 is a multiple of both 3 and 5, therefore, it is a common multiple of 3 and 5.

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