12 labourers can complete a peice of work in 72 days.How many men will be required to do the same work in 54 days?​

The equation of the line in point-slope form is:

y - 8 = 3(x + 6) or y - 2 = 3(x + 8).

The equation, y - b = m(x - a), represents the equation of a line in point-slope form, where:

Given two points, (-8, 2) and (-6, 8), find the slope (m) of the line that passes through the two points:

Slope (m) = change in y / change in x = (8 - 2)/(-6 -(-8))

Slope (m) = 6/2

Slope (m) = 3

Substitute m = 3 and (-6, 8) into y - b = m(x - a):

y - 8 = 3(x - (-6))

y - 8 = 3(x + 6)

Or substitute m = 3 and (-8, 2) into y - b = m(x - a):

y - 2 = 3(x - (-8))

y - 2 = 3(x + 8)

Therefore, the equation of the line in point-slope form is:

y - 8 = 3(x + 6) or y - 2 = 3(x + 8).

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Answer:

the last one I think

Step-by-step explanation:

The vertex form of the quadratic equation is:

\(g(x) = (x + 1)^2 - 2\)

And the graph can be seen at the end

For a quadratic equation with leading coefficient a, and with vertex (h, k), the vertex form is:

\(y = a*(x - h)^2 + k\)

Here we have:

\(g(x) = x^2 + 2x - 1\)

First, we need to find the vertex. by using the known formula we get:

\(h = -2/(2*1) = -1\)

To get the value of k, we need to evaluate g(x) in x = -1.

\(g(-1) = (-1)^2 + 2*-1 - 1 = 1 - 2 - 1 = -2\)

So the vertex is (-1, -2), which means that the vertex form is:

\(g(x) = (x + 1)^2 - 2\)

And its graph can be seen below.

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Answer:

1. Distribute

2. Combine like terms

3. Get all variable on one side and everything else on the other (Inverse operations: Add / subtract)

4. Solve for one of your variables (inverse operation: Multiple / Division

5. Simplify

Answer:

x=65

Each NBA team can have a maximum of 15 players, 13 of which can be active each game.

65/5=13 so it's safe to say tgat 5 team can be crated

Step-by-step explanation:

x/5-2=11-->transposing x/5=11+2---->x=13×5--->x=65

verification 65/5-2=11--->13-2=11

Answer: X=1.1

Step-by-step explanation: Have a brilliant day- Lily (^_^)

Answer:

58

Step-by-step explanation:

261-203= 58

simple

The explicit formula for f^-1 is (x-3)^(1/4) and this is obtained by switching the roles of x and y and solving for y in terms of x.

To find the inverse function of f(x)=x^4+3, we need to switch the roles of x and y, and solve for y.
Let y = x^4+3
Subtract 3 from both sides to get:
y - 3 = x^4
Take the fourth root of both sides to isolate x:
(x^4)^(1/4) = (y-3)^(1/4)
Simplify:
x = (y-3)^(1/4)
So the inverse function of f(x) is:
f^-1 (x) = (x-3)^(1/4)
This is the explicit formula for the inverse function of f(x).
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Answer:

1,628.89

Step-by-step explanation:

The logical expression that represents the statement "Despite the fact that it is sunny today, I will bring my umbrella" is (B) s∧m.

The logical expression that represents the statement "Despite the fact that it is sunny today, I will bring my umbrella" is (B) s∧m. This is because the conjunction "and" (∧) is used to connect two statements that must both be true in order for the overall statement to be true. In this case, both s (It is sunny today) and m (I will bring my umbrella) must be true in order for the statement to be true.

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(B) s∧m, which represents the logical expression for "Despite the fact that it is sunny today, I will bring my umbrella."


The logical operator ∧ (conjunction) connects two propositions and represents the idea of "and." Therefore, s∧m means "It is sunny today and I will bring my umbrella." This is the logical expression that represents the statement given in the question.


To summarize, the correct answer is (B) s∧m, which represents the logical expression for "Despite the fact that it is sunny today, I will bring my umbrella."
Main Answer: The correct logical expression is (B) s∧m.

In the given statement, "Despite the fact that it is sunny today, I will bring my umbrella," both propositions s (It is sunny today) and m (I will bring my umbrella) are true at the same time. The logical expression that represents this is s∧m, which means "s AND m" are both true.

The statement "Despite the fact that it is sunny today, I will bring my umbrella" is best represented by the logical expression (B) s∧m, as it shows that both s and m are true.

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Answer:

\(\textsf{(C)} \quad -\dfrac{1}{4}\)

Step-by-step explanation:

To differentiate an equation that contains a mixture of x and y terms, use implicit differentiation.

To find dy/dx for x² + 3y² = 8 + 2xy, differentiate each term with respect to x.

Begin by placing d/dx in front of each term of the equation:

\(\dfrac{\text{d}}{\text{d}x}x^2+\dfrac{\text{d}}{\text{d}x}3y^2=\dfrac{\text{d}}{\text{d}x}8+\dfrac{\text{d}}{\text{d}x}2xy\)

Differentiate the terms in x only (and constant terms):

\(\implies 2x+\dfrac{\text{d}}{\text{d}x}3y^2=0+\dfrac{\text{d}}{\text{d}x}2xy\)

Use the chain rule to differentiate terms in y only.

In practice, this means differentiate with respect to y, and place dy/dx at the end:

\(\implies 2x+6y\dfrac{\text{d}y}{\text{d}x}=0+\dfrac{\text{d}}{\text{d}x}2xy\)

\(\implies 2x+6y\dfrac{\text{d}y}{\text{d}x}=\dfrac{\text{d}}{\text{d}x}2xy\)

Use the product rule to differentiate the term in x and y.

\(\textsf{Let}\;u(x)=2x \implies u'(x)=2\)

\(\textsf{Let}\;v(x)=y \implies v'(x)= 1\dfrac{\text{d}y}{\text{d}x}\)

Therefore:

\(\implies \dfrac{\text{d}}{\text{d}x}[u(x) \cdot v(x)]=u(x) \cdot v'(x)+v(x) \cdot u'(x)\)

\(\implies \dfrac{\text{d}}{\text{d}x}2xy=2x \cdot 1 \dfrac{\text{d}y}{\text{d}x}+y\cdot 2\)

\(\implies \dfrac{\text{d}}{\text{d}x}2xy=2x \dfrac{\text{d}y}{\text{d}x}+2y\)

So the final differentiated equation is:

\(\implies 2x+6y\dfrac{\text{d}y}{\text{d}x}=2x \dfrac{\text{d}y}{\text{d}x}+2y\)

Rearrange the resulting equation to make dy/dx the subject:

\(\implies 2x+6y\dfrac{\text{d}y}{\text{d}x}=2x \dfrac{\text{d}y}{\text{d}x}+2y\)

\(\implies 6y\dfrac{\text{d}y}{\text{d}x}-2x \dfrac{\text{d}y}{\text{d}x}=-2x+2y\)

\(\implies \dfrac{\text{d}y}{\text{d}x}(-2x+6y)=-2x+2y\)

\(\implies \dfrac{\text{d}y}{\text{d}x}=\dfrac{-2x+2y}{-2x+6y}\)

\(\implies \dfrac{\text{d}y}{\text{d}x}=\dfrac{2(-x+y)}{2(-x+3y)}\)

\(\implies \dfrac{\text{d}y}{\text{d}x}=\dfrac{-x+y}{-x+3y}\)

To find d²y/dx², differentiate again using the quotient rule and implicit differentiation.

\(\textsf{Let}\;u(x)=-x+y \implies u'(x)=-1+1\dfrac{\text{d}y}{\text{d}x}\)

\(\textsf{Let}\;v(x)=-x+3y \implies v'(x)=-1+3\dfrac{\text{d}y}{\text{d}x}\)

Therefore:

\(\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{\text{d}}{\text{d}x}\left[\dfrac{u(x)}{v(x)}\right]=\dfrac{v(x)\cdot u'(x)-u(x) \cdot v'(x)}{[v(x)]^2}\)

\(\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{\text{d}}{\text{d}x}\left[\dfrac{-x+y}{-x+3y}\right]=\dfrac{(-x+3y) \cdot \left(-1+1\dfrac{\text{d}y}{\text{d}x}\right)-(-x+y) \cdot \left(-1+3\dfrac{\text{d}y}{\text{d}x}\right)}{(-x+3y)^2}\)

\(\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{(3y-x)\left(\dfrac{\text{d}y}{\text{d}x}-1\right)-(y-x)\left(3\dfrac{\text{d}y}{\text{d}x}-1\right)}{(-x+3y)^2}\)

\(\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{\left(3y\dfrac{\text{d}y}{\text{d}x}-3y-x\dfrac{\text{d}y}{\text{d}x}+x\right)-\left(3y\dfrac{\text{d}y}{\text{d}x}-y-3x\dfrac{\text{d}y}{\text{d}x}+x\right)}{(-x+3y)^2}\)

\(\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{-2y+2x\dfrac{\text{d}y}{\text{d}x}}{(-x+3y)^2}\)

Substitute in dy/dx:

\(\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{-2y+2x\left(\dfrac{-x+y}{-x+3y}\right)}{(-x+3y)^2}\)

To find d²y/dx² at (2, 2), substitute x = 2 and y = 2 into the second derivative:

\(\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{-2(2)+2(2)\left(\dfrac{-2+2}{-2+3(2)}\right)}{(-2+3(2))^2}\)

\(\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{-4+4\left(\dfrac{0}{4}\right)}{(4)^2}\)

\(\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{-4}{16}\)

\(\implies \dfrac{\text{d}^2y}{\text{d}x^2}=-\dfrac{1}{4}\)

The lengths 6, 8 and 15 cannot represent the sides of the triangle. The solution has been obtained by using the triangle inequality theorem.

What is the triangle inequality theorem?

The triangle inequality theorem outlines the relationships between a triangle's three sides. According to this theorem, the lengths of any triangle's two shorter sides add up to be longer than the triangle's third side. In other terms, this theorem states that the shortest path between any two places is always a straight line.

We are given the lengths to be 6, 8 and 15.

Using the triangle inequality theorem,

⇒ 6 + 15 > 8

⇒ 21 > 8

Also,

⇒ 8 + 15 > 6

⇒ 23 > 8

Also,

⇒ 6 + 8 > 15

But,

⇒ 14 < 15

Hence, the lengths 6, 8 and 15 cannot represent the sides of the triangle.

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A florist will used a several different types of flowers to make bouquet. Lilies 8, roses 1, and daffodils 13. Then, the probability of randomly selecting a lily from the bouquet is 4/11.

The total number of flowers in the bouquet is;

Total number of flowers = lilies + roses + daffodils = 8 + 1 + 13 = 22

The probability of randomly selecting a lily is the number of lilies in the bouquet divided by the total number of flowers;

P(lily) = number of lilies / total number of flowers = 8 / 22

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common factor, which is 2:

P(lily) = 4 / 11

Therefore, P(lily) =  4 / 11

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The value of x in the triangle is 2√13.

We have,

There are two similar triangles.

So,

The ratio of the corresponding sides is equal.

Now,

4/x = x/(9 + 4)

4/x = x / 13

4 x 13 = x²

x² = 4 x 13

x = √(4 x 13)

x = 2√13

Thus,

The value of x in the triangle is 2√13.

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Answer:

100°

Step-by-step explanation:

The sum of any polygon angles add to  ( n-2)(180)   where n is the number of sides ......for a quadrlateral    ( 4-2)(180 ) = 360

60 + 90 + 110 +  x = 360    shows    x = 100 °

Answer:

sin75^°-sin15^°=1/√2

Answer:

y = 85

Step-by-step explanation:

6x - 25 = 4x + 15

(Subtract 4x from both sides)

2x - 25 = 15

(Add 25 to both sides)

2x = 40

(Divide 40 by 2)

x = 20

6(20) - 25 = 4(20) + 15

(Solve)

95 + 95 = 190

(Subtract 360 from 190)

360 - 190 = 170

(Divide 170 by 2)

85

y = 85

By following these steps, you should be able to calculate the concentrations of [H3PO4], [H2PO4-1], [HPO4-2], and [PO4-3], as well as the concentration and KSP of [Ca3(PO4)2], and solve for Ka1, Ka2, and Ka3.

To solve for the concentration of [H3PO4], [H2PO4-1], [HPO4-2], and [PO4-3], we need to consider the acid dissociation of phosphoric acid (H3PO4). Phosphoric acid has three dissociation constants (Ka1, Ka2, and Ka3) corresponding to the three hydrogen ions it can release.

1. We start by writing the dissociation reactions for each step:

H3PO4 ⇌ H+ + H2PO4-
H2PO4- ⇌ H+ + HPO4-2
HPO4-2 ⇌ H+ + PO4-3

2. We'll assume that initially, the concentration of [H3PO4] is 150 M (as stated in the question). Since we have a pH of 8, we can calculate the [H+] using the equation pH = -log[H+]. In this case, the [H+] concentration is 10^-8 M.

3. Now, we'll use the equilibrium expression for each dissociation reaction to calculate the concentrations of [H2PO4-1], [HPO4-2], and [PO4-3].

For the reaction H3PO4 ⇌ H+ + H2PO4-, the equilibrium constant (Ka1) is given by [H+][H2PO4-] / [H3PO4]. Since we know [H3PO4] = 150 M and [H+] = 10^-8 M, we can rearrange the equation to solve for [H2PO4-]. Substitute the given values to find the concentration of [H2PO4-1].

Similarly, for the reactions H2PO4- ⇌ H+ + HPO4-2 and HPO4-2 ⇌ H+ + PO4-3, we can calculate the concentrations of [HPO4-2] and [PO4-3] using their respective equilibrium expressions.

4. Next, we can calculate the concentration and KSP of [Ca3(PO4)2] using the solubility product constant (KSP). The balanced equation for the dissolution of [Ca3(PO4)2] is:

3Ca3(PO4)2 ⇌ 9Ca2+ + 6PO4-3

Since [PO4-3] is calculated in the previous step, we can multiply it by 6 to get the concentration of [Ca2+] ions. The concentration of [Ca2+] is then used to calculate the KSP using the expression:

KSP = [Ca2+]^9 * [PO4-3]^6

5. Finally, we solve for Ka1, Ka2, and Ka3. The Ka values represent the equilibrium constants for each acid dissociation reaction.

Using the concentrations of [H+], [H2PO4-1], [HPO4-2], and [PO4-3] obtained earlier, we can calculate Ka1, Ka2, and Ka3 using the equilibrium expressions for the respective reactions.

Remember to substitute the correct concentrations into each equation to find the Ka values.

By following these steps, you should be able to calculate the concentrations of [H3PO4], [H2PO4-1], [HPO4-2], and [PO4-3], as well as the concentration and KSP of [Ca3(PO4)2], and solve for Ka1, Ka2, and Ka3.

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The concentrations of [H₃PO₄], [H₂PO₄-1], [HPO₄-2], and [PO₄-3], as well as the concentration and KSP of [Ca₃(PO₄)₂], and solve for Ka1, Ka2, and Ka3.

By following these steps, you should be able to calculate the concentrations of [H₃PO₄], [H₂PO₄-1], [HPO₄-2], and [PO₄-3], as well as the concentration and KSP of [Ca₃(PO₄)₂], and solve for Ka1, Ka2, and Ka3.

To solve for the concentration of [H₃PO₄], [H₂PO₄-1], [HPO₄-2], and [PO₄-3], we need to consider the acid dissociation of phosphoric acid (H₃PO₄).

Phosphoric acid has three dissociation constants (Ka1, Ka2, and Ka3) corresponding to the three hydrogen ions it can release.

1. We start by writing the dissociation reactions for each step:

H₃PO₄ ⇌ H+ + H₂PO₄-

H₂PO₄- ⇌ H+ + HPO₄-2

HPO₄-2 ⇌ H+ + PO₄-3

2. We'll assume that initially, the concentration of [H₃PO₄] is 150 M (as stated in the question). Since we have a pH of 8, we can calculate the [H⁺] using the equation pH = -log[H⁺]. In this case, the [H⁺] concentration is 10⁻⁸ M.

3. Now, we'll use the equilibrium expression for each dissociation reaction to calculate the concentrations of [H₂PO₄-1], [HPO₄-2], and [PO₄-3].

For the reaction H₃PO₄ ⇌ H+ + H₂PO₄-, the equilibrium constant (Ka1) is given by [H⁺][H₂PO₄-] / [H₃PO₄]. Since we know [H₃PO₄] = 150 M and [H⁺] = 10⁻⁸ M, we can rearrange the equation to solve for [H₂PO₄-]. Substitute the given values to find the concentration of [H₂PO₄-1].

Similarly, for the reactions H₂PO₄- ⇌ H+ + HPO₄-2 and HPO₄-2 ⇌ H+ + PO4-3, we can calculate the concentrations of [HPO₄-2] and [PO₄-3] using their respective equilibrium expressions.

4. Next, we can calculate the concentration and KSP of [Ca₃(PO₄)₂] using the solubility product constant (KSP). The balanced equation for the dissolution of [Ca₃(PO₄)₂] is:

3Ca₃(PO₄)₂ ⇌ 9Ca₂+ + 6PO₄-3

Since [PO₄-3] is calculated in the previous step, we can multiply it by 6 to get the concentration of [Ca²⁺] ions. The concentration of [Ca²⁺] is then used to calculate the KSP using the expression:

KSP = [Ca²⁺]⁹ * [PO₄-3]⁶

5. Finally, we solve for Ka1, Ka2, and Ka3. The Ka values represent the equilibrium constants for each acid dissociation reaction.

Using the concentrations of [H⁺], [H₂PO₄-1], [HPO₄-2], and [PO₄-3] obtained earlier, we can calculate Ka1, Ka2, and Ka3 using the equilibrium expressions for the respective reactions.

Remember to substitute the correct concentrations into each equation to find the Ka values.

By following these steps, you should be able to calculate the concentrations of [H₃PO₄], [H₂PO₄-1], [HPO₄-2], and [PO₄-3], as well as the concentration and KSP of [Ca₃(PO₄)₂], and solve for Ka1, Ka2, and Ka3.

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Answer:

x= 6.25

Step-by-step explanation:

The hypotenuse of triangle JKL is taken as 96, in the similar way the hypotenuse of triangle JTU is 36. So x can be found by first finding the value of JU, using the Pythagoras theorem.

\(a^{2} = b^{2} + c^{2}\)

In this shape the formula will be:

\(TU^{2} = JT^{2} + JU^{2}\)

Substitute the given values in the shape into the formula.

TU = 34

JT = 27

TU = Lets take 'y'

\(34^{2} = 27^{2} + y^{2}\)

1156 = 729 + \(y^{2}\)

\(y^{2}\) = 1156 - 729

\(y^{2}\) = 427

y = \(\sqrt{427}\) = 20.7 ( rounded to 1 decimal place) = 21 (rounded to whole number).

So the original value of JU is '21', but we have to find the value of 'x'. So the expression '-4 + 4x' is equal to 21. This can be written as:

-4 + 4x = 21

4x = 21 + 4

4x = 25

∴ x = \(\frac{25}{4}\) = 6.25

x= 6.25

The solution to the equation \((\sum\limits^{\infty}_{i=1} \frac{1}{2^i}) + (\sum\limits^{\infty}_{i=1} \frac{9}{10^i}) + \sin^2(\theta) + \cos^2(\theta) + e^2\) is 10.3891

From the question, we have the following parameters that can be used in our computation:

\((\sum\limits^{\infty}_{i=1} \frac{1}{2^i}) + (\sum\limits^{\infty}_{i=1} \frac{9}{10^i}) + \sin^2(\theta) + \cos^2(\theta) + e^2\)

Using the following trigonometry ratio

sin²(x) + cos²(x) = 1

We have

\((\sum\limits^{\infty}_{i=1} \frac{1}{2^i}) + (\sum\limits^{\infty}_{i=1} \frac{9}{10^i}) + \sin^2(\theta) + \cos^2(\theta) + e^2 = (\sum\limits^{\infty}_{i=1} \frac{1}{2^i}) + (\sum\limits^{\infty}_{i=1} \frac{9}{10^i}) + 1 + e^2\)

The sum to infinity of a geometric series is

S = a/(1 - r)

So, we have

\((\sum\limits^{\infty}_{i=1} \frac{1}{2^i}) + (\sum\limits^{\infty}_{i=1} \frac{9}{10^i}) + \sin^2(\theta) + \cos^2(\theta) + e^2 = \frac{1/2}{1 - 1/2} + \frac{9/10}{1 - 1/10} + 1 + e^2\)

So, we have

\((\sum\limits^{\infty}_{i=1} \frac{1}{2^i}) + (\sum\limits^{\infty}_{i=1} \frac{9}{10^i}) + \sin^2(\theta) + \cos^2(\theta) + e^2 = 1 + 1 + 1 + e^2\)

Evaluate the sum

\((\sum\limits^{\infty}_{i=1} \frac{1}{2^i}) + (\sum\limits^{\infty}_{i=1} \frac{9}{10^i}) + \sin^2(\theta) + \cos^2(\theta) + e^2 = 3 + e^2\)

This gives

\((\sum\limits^{\infty}_{i=1} \frac{1}{2^i}) + (\sum\limits^{\infty}_{i=1} \frac{9}{10^i}) + \sin^2(\theta) + \cos^2(\theta) + e^2 = 10.3891\)

Hence, the solution to the equation is 10.3891

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P(not having a computer | not connected to the internet) = P(not having a computer and not connected to the internet) / P(not connected to the internet)

To solve the given question, we need to apply the probability formula. Probability can be defined as the number of favorable outcomes divided by the number of total outcomes. In this case, we have to determine the probability that the household does not have a computer since they are not connected to the internet.

Let's assume that the probability of a household not having a computer is P(not having a computer). Similarly, the probability of households not connected to the internet is denoted by P(not connected to the internet). We need to calculate the probability of not having a computer among households not connected to the internet, which can be represented as P(not having a computer | not connected to the internet).

The formula used to calculate conditional probability is P(A|B) = P(A and B) / P(B), where A and B represent two events. Here, A represents not having a computer, and B represents not being connected to the internet. Therefore, we can use the formula as follows:

P(not having a computer | not connected to the internet) = P(not having a computer and not connected to the internet) / P(not connected to the internet)

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Answer:

12 boxes of bookmarks and 8 packs of dust covers.

Step-by-step explanation:

Shelley wants to have at least 250 bookmarks and dust covers for her 250 books.

One box of bookmarks contain 24 bookmarks.

One pack of dust cover contain 36 dust covers.

Shelley buys same number of bookmarks and dust covers.

So, we need to find out LCM(Least Common Multiple) of 24 and 36 to come to a common number that can be bought.

LCM of 24 and 36 is 72.

But, at least 250 bookmarks or dust covers have to be bought. So, we need to find a multiple of 72 which is greater than or equal to 250.

Let us consider the Table of 72:

72, 144, 216, 288, 360 .....

For having enough count of bookmarks and dust covers, at least 288 bookmarks and dust covers have to be bought.

Number of boxes of bookmarks bought = \(\frac{288}{24} \Rightarrow 12\)

Number of packs of dust covers bought = \(\frac{288}{36} \Rightarrow 8\)

12 boxes of bookmarks and 8 packs of dust covers.

Answer:

a) b ≈ 1.13

Step-by-step explanation:

The rate of auto thefts triples every 9 months.

(a) Determine, to two decimal places, the base b for an exponential model y = Ab^t of the rate of auto thefts as a function of time in months.

A = rate of auto thefts when t = 0

We are told the rate of auto thefts triple every 9 months. Hence,

3A = Ab⁹

We divide both sides by A

3A/A = Ab⁹/A

3 = b⁹

Raise both to the power of 1/9

3^⅑ = b⁹ × ⅑

b = 1.1298309639

Approximately to two decimal places

b ≈ 1.13

(b) Find the doubling time to the nearest tenth of a month.

y = Ab^t

y = 2A, b = 1.13

2A = A(1.13)^t

Divide both sides by A

2A/A = A(1.13)^t/A

2 = (1.13)^t

t = In 2/In 1.13

t = 0.6931471806/0.1222176327

t = 5.6714171702

Approximately to the nearest tenth of a month, the doubling time 5.7 months

Your friend will see you again on a Friday, 45 days from Tuesday.

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The sector is a part of the circle. Then the area of the sector will be 161 square centimeters.

It is a locus of a point drawn equidistant from the center. The distance from the center to the circumference is called the radius of the circle.

The central angle of π/3 radians and a radius of 12.4 m.

Then the area of the sector will be given as

\(\rm Area\ of\ sector = \dfrac{\theta}{2\pi} \ \pi r^2\\\\\\Area\ of\ sector = \dfrac{\pi / 3}{\pi } *\pi * 12.4^2 \\\\\\Area\ of\ sector = 161.01 \approx 161\)

The area of the sector is 161 square m.

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The to your question is that we can use the normal distribution to estimate the probability that the product will last between 16.536 and 8.054 years.


In this case, we want to calculate the probability for x = 16.536 and x = 8.054. The mean (μ) is 14.242 years, and the standard deviation (σ) is 3.978 years.

Using the formula, we can calculate the z-scores for both values:
For x = 16.536: z = (16.536 - 14.242) / 3.978
For x = 8.054: z = (8.054 - 14.242) / 3.978

Once we have the z-scores, we can look up the corresponding probabilities in the standard normal distribution table or use a calculator. Subtracting the probability for the lower z-score from the probability for the higher z-score will give us the probability that the product will last between 16.536 and 8.054 years.

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ఊhe directional derivative of f at the point (1, 2, 2) in the direction of the vector v = (1/5√2)i + (1/5√2)j is 2√2.

To find the directional derivative of the function f(x, y, z) = 2z^2x + y^3 at the point (1, 2, 2) in the direction of the vector v = (1/5√2)i + (1/5√2)j, we can use the formula for the directional derivative:

D_v(f) = ∇f · v

where ∇f is the gradient of f.

Taking the partial derivatives of f with respect to each variable, we have:

∂f/∂x = 2z^2

∂f/∂y = 3y^2

∂f/∂z = 4xz

Evaluating these partial derivatives at the point (1, 2, 2), we get:

∂f/∂x = 2(2)^2 = 8

∂f/∂y = 3(2)^2 = 12

∂f/∂z = 4(1)(2) = 8

Therefore, the gradient ∇f at (1, 2, 2) is given by ∇f = 8i + 12j + 8k.

Substituting the values into the directional derivative formula, we have:

D_v(f) = ∇f · v = (8i + 12j + 8k) · (1/5√2)i + (1/5√2)j

= 8(1/5√2) + 12(1/5√2) + 8(0)

= (8/5√2) + (12/5√2)

= (8 + 12)/(5√2)

= 20/(5√2)

= 4/√2

= 4√2/2

= 2√2

Hence, the directional derivative of f at the point (1, 2, 2) in the direction of the vector v = (1/5√2)i + (1/5√2)j is 2√2.

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The Law of Sines says that for a given triangle, the ratio between the sine of an angle and the length of the opposite side is the same for all three angles in a triangle. the sine values of all three internal angles will be the samethe sine values of all three internal angles will be proportional to each otherthe ratio between the sine of the angle and the opposite side will match for all three internal anglesthe sine values of all three angles will be different

In a triangle, we have three angles and three sides. The Law of Sines says that for any triangle, the ratio between the sine of an angle and the length of the opposite side will be the same for all three angles. In other words, the sine values of each angle are proportional to the length of the opposite side.

To put this in mathematical terms, let's consider a triangle with angles A, B, and C, and sides a, b, and c respectively. The Law of Sines can be written as:

sin(A)/a = sin(B)/b = sin(C)/c

This means that if we know the length of any two sides and the measure of the angle opposite one of those sides, we can use the Law of Sines to find the measure of the other angles and sides.

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Answer:

y^{n+4}

Step-by-step explanation:

Answer:

Step-by-step explanation:

\(\frac{y^n *y^{\frac{5}{4} } }{y^{\frac{1}{4} } } =y^n*y^{\frac{5}{4} } \times y^{-\frac{1}{4} } =y^{n+\frac{5}{4} -\frac{1}{4} } \\=y^{n+\frac{5-1}{4} } \\=y^{n+\frac{4}{4} } \\=y^{n+1}\)

Given :

The straight line depreciation equation for  a motorcycle is y = -2,150x + 17,200.

To Find :

a. What is the original price of the  motorcycle? $17,200.00  ?

b. How much value does the motorcycle  lose per year?

c. How many years will it take for the  motorcycle to totally depreciate?

Solution :

y = -2,150x + 17,200.

Here, y is price and x is unit of time in years.

a)

We know, original price is given when x =0.

So, y = 0 + 17,200

y = $17,200.00

b)

Value motorcycle lose per year is coefficient of x i.e $-2,150.00

c)

Form total depreciate :

y = 0

0 = -2,150x + 17,200

x = 17,200÷2,150

x = 8 years.

Hence, this is the required solution.

ANSWER and EXPLANATION

a) To simplify the radical, we have to:

1) First, express the number inside the radical as a product of a perfect square and a number that is not a perfect square.

2) Next, we separate the numbers into two radicals and find the square root of the perfect square.

3) Finally, we find the product of the number and the remaining radical.

B) Following the steps above:

STEP 1

STEP 2

STEP 3

A differential equation's existence and uniqueness in mathematics refers to the existence of a single, clearly defined solution that meets a certain set of requirements.

A differential equation is a type of mathematical equation that links a number of known functions or variables to an unknown function and its derivatives. If a differential equation has existence and uniqueness of solution, it means that there is only one function that satisfies the equation and corresponds to the given conditions for a given set of beginning circumstances.

The terms and structure of the differential equation, such as linearity and initial conditions, decide whether or not a solution exists. The Piccard-Lipschitz theorem, which asserts that if a function and its derivatives are locally Lipschitz continuous, then the solution to the differential equation is unique in a neighbourhood of the initial conditions, frequently ensures the uniqueness of the solution.

In conclusion, the existence and uniqueness of a solution in differential equations is a crucial idea in mathematical modelling and aids in making sure that the solutions to a given problem are clear-cut and consistent with the underlying conditions.

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