Sarina borrowed $5,800 at a simple interest rate of 7½% per year. After a certain number of years she had paid $1,305 in interest altogether. How many years was that?

The value of the expression \(\sqrt[4]{a}^3\) when evaluated is \(a^\frac34\)

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

\(\sqrt[4]{a}^3\)

Consider an expression expressed as xⁿ

The expression can be expanded as

xⁿ = x * x * x .... in n places

Using the above as a guide, we have the following:

\(\sqrt[4]{a}^3 = \sqrt[4]{a} * \sqrt[4]{a} * \sqrt[4]{a}\)

Apply the exponent rule of indices

\(\sqrt[4]{a}^3 = a^\frac14 * a^\frac14 * a^\frac14\)

Apply the power rule of indices

\(\sqrt[4]{a}^3 = a^\frac34\)

Hence, the expression \(\sqrt[4]{a}^3\) when evaluated is \(a^\frac34\)

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Answer:

Step-by-step explanation:

Find attached the second one

The tables that show a valid probability density function, are tables 1 to 3

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

Table of values

For a table to show a valid probability density function, the following must be true

Sum of P(X = x) = 1

Using the above as a guide, we have the following:

Table 1: Sum = 0.37  + 0.06 + 0.01 + 0.56 = 1

Table 2: Sum = 3/8 + 3/8 + 1/4 = 1

Table 3: Sum = 1/8 + 1/8 + 1/8 + 3/8 + 1/4 = 1

Table 4: Sum = 0.03 + 0.01 + 0.61 + 0.31 + 0.21 = 1.17

Table 5: Sum = 1/10 + 3/10 + 4/5 + 1/5 = 1.4

Hence, the tables are tables 1 to 3

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Answer:

5, 6

Step-by-step explanation:

Answer:

Los números son:

6 y 5

Step-by-step explanation:

Planteamiento:

a * b = 30

a + b = 11

Desarrollo:

De la segunda ecuación del planteamiento:

a = 11-b    

Sustituyendo esta última ecuación en la primer ecuación del planteamiento:

(11-b)*b = 30

11*b + b*-b = 30

11b - b² - 30 = 0

-b² + 11b - 30 = 0

b = {-11±√((11²)-(4*-1*-30))} / (2*-1)

b = {-11±√(121-120)} /-2

b = {-11±√1} / -2

b = {-11±1} / -2

b₁ = {-11-1} / -2 = -12/-2 = 6

b₂ = {-11+1} / -2 = -10/-2 = 5

Comprobación:

6*5 = 30

6+5 = 11

Answer:

y-coordinate is decreasing at the rate of \(\dfrac{1}{2}\) unit/sec.

Step-by-step explanation:

Given that:

The curve of the particle \(x^2y = 1\)

Then:

\(y = \dfrac{1}{x^2}\)

Taking the differential of y with respect to t

\(\dfrac{dy}{dt}= \dfrac{dx^{-2}}{dx} * \dfrac{dx}{dt}\)

\(= -2x^{-3} \dfrac{dx}{dt}\)

At (2, 1/4)

\(\dfrac{dx}{dt} = 2\)

This implies that:

\(\implies \dfrac{dy}{dt} = -\dfrac{2}{8}(2)\)

\(\dfrac{dy}{dt} = -\dfrac{1}{2} \ \ unit/sec\)

Thus, y-coordinate is decreasing at the rate of \(\dfrac{1}{2}\) unit/sec.

Let's check the significant figures one by one

\(\\ \sf\longmapsto 1,230=3\)

\(\\ \sf\longmapsto 155.6=4\)

\(\\ \sf\longmapsto 200=1\)

\(\\ \sf\longmapsto 1245=4\)

\(\\ \sf\longmapsto 8.88=3\)

\(\\ \sf\longmapsto 22=2\)

\(\\ \sf\longmapsto 136=3\)

\(\\ \sf\longmapsto 8.880=3\)

\(\\ \sf\longmapsto 0.01=1\)

\(\\ \sf\longmapsto 20=1\)

One Significant figure - 200, 0.01, 20

Two Significant figures- 22, 20.

Three Significant figures- 1,230, 8.88, 136

Four Significant figures- 155.6, 1,245, 8.880

PLATO

Step-by-step explanation:

\(2y + 106 = 180\)

\(y = 106 \div 2 = 53\)

\(y = 53\)

\(4z + 16 = 180\)

\(4z = 180 - 16\)

\(z = 164 \div 4 = 41\)

and the answer for x is 106

The required answers are 1) \($$A = x^2 + 28x + 192$$\) 2) 300000 3) \($$x^2 + 28x + 192 = (x + 14 - 2\sqrt{19})(x + 14 + 2\sqrt{19})$$\).

area of the land purchased is given as 480m², and the dimensions of the land are (x+12)mx(x+16)m. Therefore, the area of the land can be expressed as:

\($$A = (x+12)(x+16)$$\)

Expanding this expression, we get:

\($$A = x^2 + 28x + 192$$\)

Hence, the area of the land purchased is given by the polynomial expression \($x^2 + 28x + 192$\).

The total budget for the expenses is Rs. 5,00,000. If Mr. Chand's daughter is ready to share 3/5 of the expenses, then the fraction of the expenses she will pay is:

\($\frac{3}{5}=\frac{x}{500000}$$\)

Simplifying this expression, we get:

\($x = \frac{3}{5}\times 500000 = 300000$$\)

Therefore, Mr. Chand's daughter will pay Rs. 3,00,000 towards the expenses.

We can solve the polynomial \($x^2 + 28x + 192$\) into different factors by using the quadratic formula:

\($x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$\)

Here, the coefficients of the polynomial are:

\($$a = 1, \quad b = 28, \quad c = 192$$\)

Substituting these values in the quadratic formula, we get:

\($x = \frac{-28 \pm \sqrt{28^2 - 4\times 1 \times 192}}{2\times 1}$$\)

Simplifying this expression, we get:

\($$x = -14 \pm 2\sqrt{19}$$\)

Therefore, the polynomial \($x^2 + 28x + 192$\) can be factored as:

\($$x^2 + 28x + 192 = (x - (-14 + 2\sqrt{19}))(x - (-14 - 2\sqrt{19}))$$\)

or

\($$x^2 + 28x + 192 = (x + 14 - 2\sqrt{19})(x + 14 + 2\sqrt{19})$$\)

So, we have factored the polynomial into two factors.

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The vertices of the feasible region is (4, 3)

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

x + y ≤ 7

x - 2y ≤ -2

x ≥ 0

y ≥ 0

Express as equations

So, we have

x + y = 7

x - 2y = -2

Subtract the equations

3y = 9

So, we have

y = 3

Next, we have

x + 3 = 7

This gives

x = 4

Hence, the vertex of the feasible region is (4, 3)

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The cost of the molding is given as follows:

$119.30.

The circumference of a circle of radius r is given by the equation presented as follows:

C = 2πr.

The parameters for this problem are given as follows:

d = 9 -> r = 4.5.

(as the radius is half the diameter)

Hence the circumference is given as follows:

C = 2 x π x 4.5

C = 28.27 ft.

The cost is of $4.22 per ft, hence the total cost is given as follows:

4.22 x 28.27 = $119.30.

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Answer:

The third side is increasing at an approximate rate of about 0.444 meters per minute.

Step-by-step explanation:

We are given a triangle with two sides having constant lengths of 13 m and 19 m. The angle between them is increasing at a rate of 2° per minute and we want to find the rate at which the third side of the triangle is increasing when the angle is 60°.

Let the angle between the two given sides be θ and let the third side be c.

Essentially, given dθ/dt = 2°/min and θ = 60°, we want to find dc/dt.

First, convert the degrees into radians:

\(\displaystyle 2^\circ \cdot \frac{\pi \text{ rad}}{180^\circ} = \frac{\pi}{90}\text{ rad}\)

Hence, dθ/dt = π/90.

From the Law of Cosines:

\(\displaystyle c^2 = a^2 + b^2 - 2ab\cos \theta\)

Since a = 13 and b = 19:

\(\displaystyle c^2 = (13)^2 + (19)^2 - 2(13)(19)\cos \theta\)

Simplify:

\(\displaystyle c^2 = 530 - 494\cos \theta\)

Take the derivative of both sides with respect to t:

\(\displaystyle \frac{d}{dt}\left[c^2\right] = \frac{d}{dt}\left[ 530 - 494\cos \theta\right]\)

Implicitly differentiate:

\(\displaystyle 2c\frac{dc}{dt} = 494\sin\theta \frac{d\theta}{dt}\)

We want to find dc/dt given that dθ/dt = π/90 and when θ = 60° or π/3. First, find c:

\(\displaystyle \begin{aligned} c &= \sqrt{530 - 494\cos \theta}\\ \\ &=\sqrt{530 -494\cos \frac{\pi}{3} \\ \\ &= \sqrt{530 - 494\left(\frac{1}{2}\right)} \\ \\&= \sqrt{283\end{aligned}\)

Substitute:

\(\displaystyle 2\left(\sqrt{283}\right) \frac{dc}{dt} = 494\sin\left(\frac{\pi}{3}\right)\left(\frac{\pi}{90}\right)\)

Solve for dc/dt:

\(\displaystyle \frac{dc}{dt} = \frac{494\sin \dfrac{\pi}{3} \cdot \dfrac{\pi}{90}}{2\sqrt{283}}\)

Evaluate. Hence:

\(\displaystyle \begin{aligned} \frac{dc}{dt} &= \frac{494\left(\dfrac{\sqrt{3}}{2} \right)\cdot \dfrac{\pi}{90}}{2\sqrt{283}}\\ \\ &= \frac{\dfrac{247\sqrt{3}\pi}{90}}{2\sqrt{283}}\\ \\ &= \frac{247\sqrt{3}\pi}{180\sqrt{283}} \\ \\ &\approx 0.444\text{ m/min}\end{aligned}\)

The third side is increasing at an approximate rate of about 0.444 meters per minute.

9514 1404 393

Answer:

  0.444 m/min

Step-by-step explanation:

I find this kind of question to be answered easily by a graphing calculator.

The length of the third side can be found using the law of cosines. If the angle of interest is C, the two given sides 'a' and 'b', then the third side is ...

  c = √(a² +b² -2ab·cos(C))

Since C is a function of time, its value in degrees can be written ...

  C = 60° +2t° . . . . . where t is in minutes, and t=0 is the time of interest

Using a=13, and b=19, the length of the third side is ...

  c(t) = √(13² +19² -2·13·19·cos(60° +2t°))

Most graphing calculators are able to compute a numerical value of the derivative of a function. Here, we use the Desmos calculator for that. (Angles are set to degrees.) It tells us the rate of change of side 'c' is ...

  0.443855627418 m/min ≈ 0.444 m/min

_____

Additional comment

At that time, the length of the third side is about 16.823 m.

__

c(t) reduces to √(530 -494cos(π/90·t +π/3))

Then the derivative is ...

  \(c'(t)=\dfrac{494\sin{\left(\dfrac{\pi}{90}t+\dfrac{\pi}{3}\right)}\cdot\dfrac{\pi}{90}}{2\sqrt{530-494\cos{\left(\dfrac{\pi}{90}t+\dfrac{\pi}{3}}}\right)}}}\\\\c'(0)=\dfrac{247\pi\sqrt{3}}{180\sqrt{283}}\approx0.443855...\ \text{m/min}\)

The sum of the first 70 terms of an arithmetic sequence with first term 14 and common difference 1/2 is 4375/2.

Sum of an arithmetic sequence is expressed as;

Sn = n/2 × [2a+(n−1)d]

Given the data in the question;

Plug in the given values.

Sn = n/2 × [2a+(n−1)d]

S₇₀ = 70/2 × [2(14) + ( 70 − 1 )1/2]

S₇₀ = 35 × [28 + (69 )1/2]

S₇₀ = 35 × [28 + 69/2]

S₇₀ = 35 × 125/2]

S₇₀ = (35 × 125)/2

S₇₀ = 4375/2

The sum of the first 70 terms of an arithmetic sequence with first term 14 and common difference 1/2 is 4375/2.

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The term that should be added to the expression to make the expression perfect square trinomial is 169/4. The expression then becomes : (y - 13/2)²

What is meant by a perfect square trinomial?

By multiplying a binomial by another binomial, perfect square trinomials—algebraic equations with three terms—are created. A number can be multiplied by itself to produce a perfect square. Algebraic expressions known as binomials are made up of simply two words, each of which is separated by either a positive (+) or a negative (-) sign. Similar to polynomials, trinomials are three-term algebraic expressions.

A perfect square trinomial expression can be created by taking the binomial equation's square. If and only if a trinomial satisfying the criterion b² = 4ac has the form ax² + bx + c, it is said to be a perfect square.

Given expression y² - 13y + ?

Comparing with the general equation

a = 1

b = -13

For perfect square trinomial

b² = 4ac

(-13)² = 4 * 1 * c

169 = 4c

c = 169/4

So the expression becomes,

y² - 13y + 169/4 = (y - 13/2)²

Therefore the term that should be added to the expression to make the expression perfect square trinomial is 169/4.

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The rate of change over the interval is the difference in the y coordinate over the difference in x-coordinate over between the endpoints of the interval.

Now, in our case

Hence, choice C is correct.

Answer:

AC = 8

Step-by-step explanation:

Area of kits is half the product of diagonals

\(Area\; of \;kite = \frac{d_1d_2}{2}\\\\36 = \frac{AC*BD}{2} \\\\36 = \frac{AC(6+3)}{2} \\\\36*2 =9AC\)

AC = 72/9

AC = 8

Answer:

x=E

Step-by-step explanation:

Step-by-step explanation:

Slope:(II)m= -2

y-1=-2(x+4)

y-1=-2x+8

+1. +1

Answer: y=-2x+9

Answer:

The null hypothesis is  \(H_o : \mu_1 = \mu_2\)

The alternative hypothesis is   \(H_a : \mu_1 < \mu_2\)

The test statistics is   \(t =  -2.65\)

Reject the null hypothesis

There is  sufficient evidence to conclude that the weight of the old aluminium package pills is less than the new aluminium pills.

Step-by-step explanation:

From the question we are told that

The data for X is  373.2 , 376.7  , 381.6 , 382.1  , 388.7 ,  384.0 ,  397.9 , 389.8 ,385.1 , 371.3 , 383.5

The data for Y is 395.5 ,  384.8 , 383.5  , 386.5  ,394.8

, 391.6  , 397.7 , 384.0 , 391.7 , 398.8

Generally the sample mean for X is mathematically represented as

          \(\= x = \frac{\sum x_i }{n}\)

=>       \(\= x = \frac{373.2 +376.7 +\cdots + 383.5}{11}\)

=>       \(\= x = 383.08 \)

Generally the sample mean for Y is mathematically represented as

          \(\= y = \frac{\sum y_i }{n}\)

=>       \(\= y = \frac{395.5 +384.8 +\cdots + 398.8}{10}\)

=>       \(\= y = 390.89 \)

Generally the standard deviation for  X is mathematically represented as

     \(\sigma_1 = \sqrt{\frac{\sum (x_i - \= x_1 )^2}{n} }\)

=>   \(\sigma_1 = \sqrt{\frac{( 373.2 - 383.08)^2 +( 376.7 - 383.08)^2 +\cdots + ( 383.5 - 383.08)^2 }{11} }\)

=> \(\sigma_1 = 7.63 \)  

Generally the standard deviation for  Y is mathematically represented as

     \(\sigma_2 = \sqrt{\frac{\sum (y_i - \= y )^2}{n} }\)

=>   \(\sigma_2 = \sqrt{\frac{( 395.5 - 390.89)^2 +( 384.8 - 390.89)^2 +\cdots + ( 398.8 - 390.89)^2 }{10} }\)

=> \(\sigma_2 = 5.82 \)  

The null hypothesis is  \(H_o : \mu_1 = \mu_2\)

The alternative hypothesis is   \(H_a : \mu_1 < \mu_2\)

Generally the test statistics is mathematically represented as

     \(t = \frac{\= x - \= y}{ \sqrt{\frac{\sigma_1^2 }{ n_1 } +\frac{\sigma_2^2 }{ n_2 } } }\)

=>   \(t = \frac{383.08 - 390.89}{ \sqrt{\frac{7.63^2 }{ 11 } +\frac{5.82^2 }{10} } }\)

=>   \(t = -2.65\)

Generally the level of significance is \(\alpha = 0.05\)

Generally the degree of freedom is mathematically represented as

      \(df = n_1 + n_2 - 2\)

=>   \(df = 11 + 10 - 2\)

=>   \(df = 19\)

Generally from the student t- distribution table the critical value of   \( \alpha \) at a df = 19 is      

    \(t_{\alpha ,df} =t_{0.05 ,19} = -2.093\)

Now comparing the critical value  obtained with test statistic calculated we see that the critical value is the region of the calculated test statistic( i.e  between  -2.65 and  2.65) hence we reject the null hypothesis

Therefore there sufficient evidence to conclude that the old aluminium package pills weight is less than the new aluminium pills

The ratio of chocolate bars More Preferred: (9 chocolate bars, 3 glasses of milk); Equally Preferred: (6 chocolate bars, 3 glasses of milk); Less Preferred: (3 chocolate bars, 2 glasses of milk) and (9 chocolate bars, 2 glasses of milk).

The ratio of chocolate bars to glasses of milk that Isabel consumes is 3-to-1. This means that whenever she has 3 chocolate bars, she also has 1 glass of milk. Currently, Isabel has 6 chocolate bars and 2 glasses of milk. The bundles listed above are compared to Isabel's current consumption. Out of the five bundles, the one with 9 chocolate bars and 3 glasses of milk is more preferred than what Isabel currently has because it has more of the ratio than Isabel's current consumption. The bundle with 6 chocolate bars and 3 glasses of milk is equally preferred because it holds the same ratio as Isabel's current consumption. The bundles with 3 chocolate bars and 2 glasses of milk and 9 chocolate bars and 2 glasses of milk are less preferred because they both have less of the ratio than Isabel's current consumption.

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Answer:my answer is 15,490,000

Step-by-step explanation: cuz that is what rounding does

Answer:

55p +75a = 2725

Step-by-step explanation:

Answer:

The answer is A. Aka 6

Step-by-step explanation:

1. Revised Net Operating Income = $66,760

2. Revised Net Operating Income  =$64,640

3. Revised Net Operating Income  =$52,

1. If the sales volume increases by 40 units:

So, New Sales = 8,700 units + 40 units = 8,740 units

and, New Contribution Margin =

= $14.00 x 8,740 units

= 122, 360

New Fixed Expenses remain the same at $55,600

Then, Revised Net Operating Income

= New Contribution Margin - New Fixed Expenses

= 122360 - 55600

= 66,760.

2. If the sales volume decreases by 40 units:

New Sales = 8,700 units - 40 units = 8,660 units

New Contribution Margin

= 14 x 8660

= 121,240

New Fixed Expenses remain the same at $55,600

Then, Revised Net Operating Income

= New Contribution Margin - New Fixed Expenses

= 65,640

3. If the sales volume is 7,700 units:

New Sales = 7,700 units

New Contribution Margin

= 14 x 7700

= 107,800

New Fixed Expenses remain the same at $55,600

Then, Revised Net Operating Income

= New Contribution Margin - New Fixed Expenses

= 52, 200

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Answer:4.8π

Step-by-step explanation:

Φ=48°

Area of circle=36π

Area of circle=π x r^2

36π=πr^2

Dividing both sides by π we get

r^2=36π/π

r^2=36

Take them square root of both sides

√(r^2)=√(36)

r=6

Area of sector=Φ/360 x π x r x r

Area of sector=48/360 x π x 6 x 6

Area of sector=(48xπx6x6)/360

Area of sector=1728π/360

Area of sector=4.8π

Answer:

− 4 y  − 7

Step-by-step explanation:

Remove parentheses.

− 4 y − 4 − 3

Subtract  3  from  − 4

− 4 y  −  7

.

Answer:

36

Step-by-step explanation:

The constant differences between the consecutive terms are 2 (a); 2 (b), -3 (c), 7 (d), 1(e), and 6(f).

In math, the constant difference can be defined as the number that defines the pattern of a sequence of numbers. This means that number that should be added or subtracted to continue with the sequence.

Due to this, to determine the constant difference it is important to observe the pattern and find out the number that should be added. For example, if the sequence is 2, 4, 6, 8, there is a difference of 2 between each of the numbers and this is the constant difference.

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