why can't a system of linear equations have exact solutions s, where s is an integer that is definitely greater than 1?

a) 30 minutes are taken to have the same amount of water.

b) Both pools have an amount of 1372.5 liters when 30 minutes have passed.

A linear function is one that produces a straight line when plotted. Generally, it is a polynomial function with a maximum degree of 1 or 0. 

Now in the given question ,

a) Physically speaking, the capacity (Q) of each pool, in liters, is equal to the product of flow rate , in liters per minute, and time (t), in minutes. Hence, we derive the following functions :

First pool,

[tex]Q_1=915+15.25t\\[/tex]           ...... (1)

Second pool,

[tex]Q_2=45.75t[/tex]                   ....... (2)

The following expression can be used to calculate how long it will take to find two pools with the same amount of water:

[tex]Q_1=Q_2[/tex]                       ........ (3)

By putting value of (1) and (2) in (3),

915 + 15.25 t = 45.75 t

30.5 t = 915

t = 915 ÷ 30.5

t = 30 minutes

30 minutes are taken to have the same amount of water.

b) By (2) and knowing that t = 30 , then we have the corresponding amount:

[tex]Q_2=45.75t\\\\Q_2=45.75*30\\\\Q_2=1372.5L[/tex]

Both pools have an amount of 1372.5 liters when 30 minutes have passed.

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The volume of an aquarium is 2071.875 cubic inches. Below, you will learn how to solve the problem.

The volume of an aquarium can be calculated by multiplying its length, width, and height. In this case, the length is 19 1/2 inches, the width is 8 1/2 inches, and the height is 12 1/2 inches.

To calculate the volume, we need to multiply these three measurements together:

Volume = Length × Width × Height
Volume = (19 1/2) × (8 1/2) × (12 1/2)
Volume = 2071.875 cubic inches

So the volume of the aquarium is 2071.875 cubic inches.

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The solution to the differential equation is:

[tex]y = \int\limits [(2e^(2x) + 2e^x - 3D^2(e^x+1)^2) / (D^3+1)^2] dx + C[/tex]

What is a differential equation?

Any equation involving the derivatives of one dependent variable with respect to another independent variable is referred to as a differential equation. Differential equations are widely used in mathematics, but they also play an important role in the sciences of medicine, chemistry, physics, and engineering.

To solve this differential equation, we need to find a function y(x) that satisfies the equation. Here's how to do it:

First, we can divide both sides of the equation by D³ + 1 to get:

y = (eˣ+1)² / (D³+1)

Next, we can take the derivative of both sides with respect to x:

dy/dx = d/dx[(eˣ+1)² / (D³+1)]

Using the quotient rule of differentiation, we get:

dy/dx = [(D³+1)(2eˣ)(eˣ+1) - (eˣ+1)²(3D²)] / (D³+1)²

Simplifying the numerator, we get:

[tex]dy/dx = [2e^{(2x)} + 2e^x - 3D^{2}(e^x+1)^2] / (D^3+1)^2[/tex]

Therefore, the solution to the differential equation is:

[tex]y = \int\limits [(2e^(2x) + 2e^x - 3D^2(e^x+1)^2) / (D^3+1)^2] dx + C[/tex]

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The minimum value of the bacterial population on the interval [2, 6] is 4, and the maximum value is 64.

To find the minimum and maximum values of the bacterial population along the interval [2, 6], we need to evaluate the function b(t) at t = 2 and t = 6, respectively, and then compare the results.

at t = 2

b(2) = 2^2 = 4, which means that after 2 days, the sample has 4 bacteria.

at t = 6

b(6) = 2^6 = 64, which means that after 6 days, the sample has 64 bacteria.

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Answer:

0.06

Step-by-step explanation:

Identify the probability to the nearest hundredth that a point chosen randomly inside the rectangle either is in the hexagon or in the circle

Area of rectangle = 26.2 * 13 =  340.6 sq inch

Area of Hexagon  = 3√3(side)²/2 = 3√3 (1.8)²/2 = 8.42 sq inch

Area of circle = π(radius)² = 3.14 * (2)² = 12.56 sq inch

Area of hexagon + area of circle = 12.56 + 8.42 = 20.98 sq inch

Probability of selecting point inside the hexagon or in the circle  = 20.98/340.6

= 0.06

As a result, the island's population will be about 1397.33 persons in 6 years.

An exponent, also called a power or index, is a mathematical operation that indicates the number of times a base number is multiplied by itself. Exponents are written as a superscript number to the right of the base number. Exponents can be positive or negative, and they can be whole numbers or fractions. A positive exponent tells us to multiply the base number by itself, while a negative exponent tells us to divide the base number into 1. Exponents can be used to simplify and solve many mathematical problems, and they are an important part of many areas of mathematics, including algebra, calculus, and geometry.

Here,

If the population of the island doubles every 34 years, we can use the exponential growth formula to determine the population after a certain period of time:

P = P0 * 2ⁿ÷³⁴

where P is the population after time t, P0 is the initial population, and t is the time elapsed in years.

We know that the current population is 1233, so P0 = 1233. We want to find the population 6 years from now, so t = 6. Plugging these values into the formula, we get:

P = 1233 * 2⁶÷³⁴

P ≈ 1397.33

Therefore, the population of the island will be approximately 1397.33 people 6 years from now.

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The limit definition of the derivative is written as [tex]f '(x) &= \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}[/tex]. Here, h is defined as (x₂ – x₁) or ∆x or the change in x.

The limit definition of the derivative is also known as the difference quotient or increment definition of the derivative.  This is a product of the input value difference, (x + h) - x, and the function value difference, f(x + h) - f(x). This can be calculated using the difference quotient formula as follows,

[tex]\begin{aligned}f '(x) &= \lim_{h \to 0}\;\text{(difference quotient)}\\f '(x) &= \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}\end{aligned}[/tex].

Here, f(x) represents (y₁), f(x+h) represents (y₂), x represents x₁, x+h represents x₂, h represents (x₂ – x₁) or ∆x or the change in x, Lim represents the slope M as h→0, and f (x+h) – f (x) – represents (y₂ – y₁).

This provides a measurement of the function's average rate of change over an interval. In other words, this provides the current rate of change.

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The mean of the data is 16.5 and the standard deviation is 7.28

The given data set is a small sample of 12 observations.

To determine the type of distribution, we can first create a histogram or a boxplot of the data.

A histogram of the data shows that the distribution is unimodal and slightly right-skewed.

Alternatively, we can calculate the skewness of the data. If the skewness is close to zero, then the data is approximately symmetric. If the skewness is positive, then the data is right-skewed. If the skewness is negative, then the data is left-skewed.

Calculating the skewness of the data set, we get:

skewness = (n / ((n - 1) * (n - 2))) * Sum[(xi - x-bar)^3 / s^3]

where n is the sample size, x-bar is the sample mean, s is the sample standard deviation, and Sum is the sum of the values in the data set.

Using this formula, we get a skewness of approximately 0.456, which indicates that the distribution is slightly right-skewed.

Based on these findings, we can conclude that the distribution of the data set is approximately normal, but slightly right-skewed.

To find the best measure of center and spread, we can calculate the sample mean and sample standard deviation, respectively.

Sample mean:

mean = (1 + 7 + 11 + 14 + 17 + 17 + 17 + 21 + 21 + 23 + 23 + 26) / 12 = 16.5

Sample standard deviation:

s = sqrt((1/11) * [(1 - 16.5)^2 + (7 - 16.5)^2 + ... + (26 - 16.5)^2]) = 7.28

Therefore, the best measure of center for this data set is the sample mean of 16.5, and the best measure of spread is the sample standard deviation of 7.28.

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Answer:

pair c has the inverse functions

The conversion value of the given quantity 58 percentage in decimals is equal to 0.58.

One percentage converted to decimals is written as :

1 percentage = ( 1 / 100 )

                      =  ( 0.01 )

Now Get the value in decimals for 58 percentage using conversion factor we have ,

= 58 percentage

= ( 58 / 100 )

= ( 0.58 ) in decimal form

Therefore, the value which we get after conversion of percentage to decimals is equal to 58 percentage = 0.58 in decimals.

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Answer:

Step-by-step explanation:

The length of each side of the square by factoring the area expression are  (4x+3).

A square's area is equal to the square of each side since all of its sides are the same length.

The following is the formula for determining a square's area:

A = L²

where L is the length of each side of square.

Given the area of a square expressed as 16x² + 24x + 9

Factorize 16x² + 24x + 9

16x² + 24x + 9

=  16x² + 12x + 12x + 9

= 4x(4x+3) + 3(4x+3)

= (4x+3)²

Hence the length of each side of the square is (4x+3).

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In the given homogeneous system in parametric vector form, where the solution set is x = x₁ x₂ x₃, x = x₃. where the value of x₁ = 5k, x₂ = 5k and x₃ = 10k.

The system is

x₁ + x₂ - 15x₃ = 0 .......(1)

2x₁ + x₂ - 15x₃ = 0 .......(2)

-x₁ + x₂ = 0

We write the system in the matrix form or AX = 0

Where A = ( 1   2   -15)                 and X = ( x₁)

                 ( 2   1    -15)                               (x₂)

                 ( -1   1      0)                               (x₃)

Now, | A | = [15] -2 [-15] -15 [2+1]

= 15 + 30 - 45

= 0

∴ Since, | A | = 0, ∴ Given system has non-trivial solution.

Let x₃ = k

∴ From equation (1), x₁ + x₂ = 15k

from equation (2), 2x₁ + x₂ = 15k × 2

4x₁ + 2x₃ = 30k - x₁  + 2x₂ = 15k

= 3x₁ = 15k

∴ x₁ = 5k

and x₂ = 15k - 2x₁ = 15k - 10k

∴ x₂ = 5k.

To find x₃ = x₁ + x₂

x₃ = 5k + 5k

∴ x₃ = 10k.

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The radius of the cone in terms of x is: r = √(x² + 14x + 49). This can be solved by using volume of cone formula.

Volume of cone (V) = 1/3πr²h, where r is the radius and h is the height of the cone.

Given:

h = 9 in.

V = 3πx² + 42πx + 147π

r = ?

Substitute

3πx² + 42πx + 147π = 1/3(π)(r²)(9)

3π(x² + 14x + 49) = (π)(r²)(3)

Divide both sides by 3π

x² + 14x + 49 = r²

Square on both sides

√(x² + 14x + 49) = r

r = √(x² + 14x + 49)

Therefore, the radius of the cone in terms of x is: r = √(x² + 14x + 49).

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Answer:

2

Step-by-step explanation:

The seventh root of a number is the number that would have to be multiplied by itself 7 times to get the original number so if you multiply 2 by itself 7 times you'd get 128

The measures of the angles of the triangle are:

Angle 1: −3 degrees
Angle 2: 10 degrees
Angle 3: −14 degrees

To solve for g and find the measures of the angles of the triangle, we need to use the fact that the sum of the angles in a triangle is 180 degrees. This means that:

−3 + (2+8) + (3−17) = 180

Simplifying the equation, we get:

−3 + 10 + (−14) = 180

−7 = 180

To solve for g, we need to isolate it on one side of the equation. To do this, we can add 7 to both sides of the equation:

g = 180 + 7

g = 187

Now, we can plug g back into the original equation to find the measures of the angles:

−3 + (2+8) + (3−17) = 187

−3 + 10 + (−14) = 187

−7 = 187

Therefore, the measures of the angles of the triangle are:

Angle 1: −3 degrees
Angle 2: 10 degrees
Angle 3: −14 degrees

I hope this helps! Let me know if you have any further questions.

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The postulate that can be used to prove that triangles ABC and DEF are similar triangles is A. AA Similarity Postulate.

According to the AA triangle similarity criterion, two triangles are considered similar if their three angles are identical to one another. Equi-angular triangles are comparable in a nutshell. As just two pairs of angles must be equal in order for the third pair to be automatically equal thanks to the angle sum property of triangles, the proper term for this criterion would then be the AAA(Angle-Angle-Angle) criterion. However, we refer to it as the AA criterion instead.

We have, triangles ABC and DEF with:

two pairs of congruent angles - ∠B ≅ ∠E and ∠C ≅ ∠F.

Therefore, the postulate that has been used to prove that the triangles ABC and DEF are similar triangles is AA Similarity Postulate.

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The complete question is:

Which theorem or postulate proves that â–³ABC and â–³DEF are similar?Select from the drop-down menu to correctly complete the statement. The two triangles are similar by the ________. A. AA Similarity Postulate B. SSS Similarity Theorem C. SAS Similarity Theorem

A subset is when all elements of any set A are present in another set B. Its working has been explained below.

Let us take two sets on random i.e., set A and set B. If we find that all elements of set A can be found in set B as well, we deem it to be a subset of set B. If we were to explain in simpler terms, we can say that set A is a part of set B.

Let us further assume that set A has the elements {1,2,3} and set {1,2,3,4,5,6}. From what we can see is that all elements in set A are also present in set B. This means that set A is a subset of set B.

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Answer:

$129,870.13 AUD

Step-by-step explanation:

To find out what we need to solve, we can use these exchange rates:

Now, we need to divide 100,000 by 0.77.

We do this because we need to figure out how many times 0.77 must be multiplied to get 100,000. Then, we take that amount, and multiply that by however many AUD's you have.

Therefore, for every $100,000 USD, there is $129,870.13 AUD.

Answer:

May you please give me the brainliest answer:

To find the value of x, we can set the two given equations equal to each other, since they both represent the same value of y:

2/3x + 8 = 1/2x + 10

We can simplify this equation by first getting rid of the fractions. To do this, we can multiply both sides of the equation by the least common multiple of the denominators, which is 6:

6(2/3x + 8) = 6(1/2x + 10)

4x + 48 = 3x + 60

Next, we can isolate the x term on one side of the equation by subtracting 3x from both sides:

4x - 3x + 48 = 60

Simplifying this gives:

x + 48 = 60

Finally, we can isolate x by subtracting 48 from both sides:

x = 60 - 48 = 12

Therefore, x = 12.

The total number of pages in Rhett's book is 183 pages.

A solution to an equation is a number that can be substituted for the variable to provide a piece of true number information.

To determine the total number of pages in Rhett's book, we need to solve for p in the equation:

p + 31 = 214

First, we can separate p on one side of the equation by subtracting 31 from both sides:

p + 31 - 31 = 214 - 31

Simplifying:

p = 183

Therefore, the total number of pages in Rhett's book is 183 pages.

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Answer:

Inscribed angle

Step-by-step explanation:

Central angles have the vertex as the center of the circle, and chords are not angles

The probability that the sample average exceeded 5100 is given as follows:

0%.

The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The parameters for this problem are given as follows:

[tex]\mu = 250, \sigma = 5000, n = 64, s = \frac{5000}{\sqrt{64}} = 625[/tex]

The probability that a sample average exceeds $5,100 is one subtracted by the p-value of Z when X = 5100, hence:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

By the Central Limit Theorem:

[tex]Z = \frac{X - \mu}{s}[/tex]

Z = (5100 - 250)/625

Z = 7.76

Z = 7.76 has a p-value of 1.

1 - 1 = 0.

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The yield of 5x2 is 10

Yield is a term used in mathematics to describe the output or result of a calculation or operation. In this case, we are looking at the yield of 5 multiplied by 2 (5x2).

When we multiply two numbers together, we are finding the yield or product of those numbers. In this case, 5 multiplied by 2 gives us a yield of 10. So, the yield of 5x2 is 10.

It's important to note that yield can be used in different contexts in mathematics, such as in finance to describe the return on an investment or in chemistry to describe the amount of product obtained from a chemical reaction. But in this case, we are simply looking at the yield of a multiplication operation.

In summary, which means that when we multiply 5 and 2 together, we get a result or output of 10.

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The area for the given geometric shape will be 166.5 inch² i.e. B.

What are geometric shapes, exactly?

Geometric forms are mathematical representations that represent the shape of everyday items. Geometric forms with boundary lines, angles, and surfaces are known as shapes. There are several 2d and 3d forms.

Shapes can also be classed based on their regularity or homogeneity. A symmetrical regular form, such as a square or circle, is common. Asymmetry can take on irregular shapes. They're also referred to as organic forms or freeform shapes. A tree, for example, has an irregular or organic shape.

In plane geometry, two-dimensional forms are flat shapes and closed figures such as circles, squares, rectangles, rhombuses, and so on. In solid geometry, the three-dimensional shapes are the cube, cuboid, and sphere.

Now,

Area of the shape will be = Area of rectangle + Area of triangle

For rectangle, length=16 inch, Breadth=9 inch

For triangle, base=6 inch, height=7.5 inch

then area of shape=l*b+1/2*b*h

=16*9+1/2*6*7.5

=144+22.5

=166.5 inch²

hence,

             The area for the given geometric shape will be 166.5 inch².

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Answer: a) To find the height of the pyramid, we can use the Pythagorean theorem. Let H be the height of the pyramid, and let M be the midpoint of QR. Then, PM is half of PQ, which is 15 cm. We can use the Pythagorean theorem to find H:

H^2 = PQ^2 - PM^2

H^2 = 30^2 - 15^2

H^2 = 675

H = sqrt(675) = 5 sqrt(3) cm

b) To find the angle that PS makes with the base QRST, we can use trigonometry. Let A be the foot of the perpendicular from P to the base QRST, and let B be the midpoint of PS. Then, we have:

tan(angle PSB) = AB / PB

Since triangle PAB is a right triangle, we can use the Pythagorean theorem to find AB:

AB^2 = AP^2 - PB^2

AB^2 = H^2 + (PQ/2)^2 - PB^2

AB^2 = (5 sqrt(3))^2 + (15/2)^2 - PB^2

AB^2 = 225/4 + 75 - PB^2

AB^2 = 375/4 - PB^2

Since triangle PBS is also a right triangle, we can use the Pythagorean theorem to find PB:

PB^2 = PS^2 - BS^2

PB^2 = (2 H)^2 - (PQ/2)^2

PB^2 = 4 (5 sqrt(3))^2 - (15/2)^2

PB^2 = 500 - 56.25

PB^2 = 443.75

Substituting these values into the equation for tan(angle PSB), we get:

tan(angle PSB) = sqrt(375/4 - 443.75) / sqrt(443.75)

tan(angle PSB) = -0.4385

Since angle PSB is in the second quadrant, we have:

angle PSB = 180 degrees + arctan(-0.4385) = 152.4 degrees (rounded to one decimal place)

c) To find the total surface area of the pyramid, including the base, we can divide the pyramid into four triangular faces and a square base. The area of each triangular face can be found using the formula:

area = (1/2) base * height

where the base is the length of one edge of the square base, and the height is the height of the pyramid. The area of the base is simply the area of the square QRST, which is (20 cm)^2 = 400 cm^2. Therefore, we have:

area of each triangular face = (1/2) (20 cm) (5 sqrt(3) cm) = 50 sqrt(3) cm^2

total surface area = 4 (50 sqrt(3) cm^2) + 400 cm^2 = 200 sqrt(3) cm^2 + 400 cm^2

total surface area = (200 + 200 sqrt(3)) cm^2 ≈ 532.4 cm^2 (rounded to one decimal place)

Step-by-step explanation:

If the  fruit seller threw some rotten apples away and the ratio of apples to oranges became 1 : 4. The fruit seller threw away 630 rotten apples.

To find the number of rotten apples that the fruit seller threw away, we will need to use the given information and set up equations to solve for the unknown variable.

Let's start by using the given information that there were 2/3 as many apples as oranges. We can set up an equation to represent this relationship:

2/3 O = A

Where O is the number of oranges and A is the number of apples.

We are also given that the fruit seller had 2520 apples and oranges in total. We can set up another equation to represent this relationship:

O + A = 2520

Now, we can use the first equation to solve for one of the variables in terms of the other. Let's solve for A in terms of O:

A = (2/3)O

Now we can substitute this expression for A into the second equation:

O + (2/3)O = 2520

Simplifying this equation gives us:

(5/3)O = 2520

Now we can solve for O:

O = (3/5)(2520) = 1512

Now that we know the number of oranges, we can use the first equation to solve for the number of apples:

A = (2/3)(1512) = 1008

So the fruit seller originally had 1008 apples and 1512 oranges.

We are also given that after throwing away some rotten apples, the ratio of apples to oranges became 1:4. We can set up an equation to represent this relationship:

(A - X)/O = 1/4

Where X is the number of rotten apples that were thrown away.

Substituting the values we found for A and O into this equation gives us:

(1008 - X)/1512 = 1/4

Cross-multiplying and simplifying gives us:

4(1008 - X) = 1512

4032 - 4X = 1512

4X = 2520

X = 630


Therefore, the answer to the question is 630.

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The solutions to the equation k³ + 6k² + 9k = 0 are given as follows:

k = 0 and k = -3.

The equation for this problem is defined as follows:

k³ + 6k² + 9k = 0.

Simplifying the common factor k, we have that:

k(k² + 6k + 9) = 0.

The first solution is of:

k = 0.

The second solution is obtained as follows:

k² + 6k + 9 = 0

Applying the perfect square trinomial:

(k + 3)² = 0

k + 3 = 0

k = -3.

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