if $5700 is invested in a savings account for which interest is compounded annually, and if the $5700 turns into $6100 in 12 years, what is the interest rate of the savings account?

Answer:

m = 3 + 2√3 or m = 3 - 2√3

Step-by-step explanation:

The given equation is:

m^2 - 6m = 3

We can rewrite the equation in standard quadratic form as:

m^2 - 6m - 3 = 0

We can use the quadratic formula to solve for m:

m = [-b ± √(b^2 - 4ac)] / 2a

where a = 1, b = -6, and c = -3.

Substituting these values, we get:

m = [-(-6) ± √((-6)^2 - 4(1)(-3))] / 2(1)

m = [6 ± √(36 + 12)] / 2

m = [6 ± √48] / 2

m = [6 ± 4√3] / 2

Simplifying the expression, we get:

m = 3 ± 2√3

Therefore, the solutions to the equation are:

m = 3 + 2√3 or m = 3 - 2√3

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Answer:

[tex]x=3+2\sqrt{3}\\\\ x=3-2\sqrt{3}[/tex]

Step-by-step explanation:

The quadratic formula is:

[tex]x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\quad\textsf{when }\:ax^2+bx+c=0[/tex]

Therefore, to solve the given equation using the quadratic formula, first subtract 3 from both sides of the equation so that it is in the required form.

[tex]m^2-6m-3=3-3[/tex]

[tex]m^2-6m-3=0[/tex]

Comparing the coefficients:

Substitute the values of a, b and c into the formula and solve for x:

[tex]\implies x=\dfrac{-(-6) \pm \sqrt{(-6)^2-4(1)(-3)}}{2(1)}[/tex]

[tex]\implies x=\dfrac{6 \pm \sqrt{36-4(-3)}}{2}[/tex]

[tex]\implies x=\dfrac{6 \pm \sqrt{36+12}}{2}[/tex]

[tex]\implies x=\dfrac{6 \pm \sqrt{48}}{2}[/tex]

Rewrite 48 as a 4² · 3:

[tex]\implies x=\dfrac{6 \pm \sqrt{4^2 \cdot 3}}{2}[/tex]

[tex]\textsf{Apply the radical rule:} \quad \sqrt{ab}=\sqrt{a}\sqrt{b}[/tex]

[tex]\implies x=\dfrac{6 \pm \sqrt{4^2} \sqrt{3}}{2}[/tex]

[tex]\textsf{Apply the radical rule:} \quad \sqrt{a^2}=a, \quad a \geq 0[/tex]

[tex]\implies x=\dfrac{6 \pm 4 \sqrt{3}}{2}[/tex]

Simplify:

[tex]\implies x=\dfrac{6}{2}\pm\dfrac{4 \sqrt{3}}{2}[/tex]

[tex]\implies x=3\pm2 \sqrt{3}[/tex]

Therefore, the values of x are:

[tex]x=3+2\sqrt{3}\\\\ x=3-2\sqrt{3}[/tex]

The value of v in terms of i and j is v = 3i + 3j

We are given that;

P₁ = (-5,3), P₂ = (-2,6)

Now,

To find the component form of a vector, we need to subtract the coordinates of the initial point from the coordinates of the terminal point3.

We have P₁ = (-5,3) and P₂ = (-2,6).

v = P₂ - P₁ = (-2 - (-5), 6 - 3) = (3, 3). We can write v in terms of i and j as v = 3i + 3j.

Therefore, by vectors the answer will be v = 3i + 3j.

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The value of the given expression in terms of i and j is -78i-12j.

Given that, v = - 8i+2j and w = -i-5j.

We need to find the value of 6w+9v.

Substitute v = - 8i+2j and w = -i-5j in 6w+9v, we get

6(-i-5j)+9(-8i+2j)

= -6i-30j-72i+18j

= -78i-12j

Therefore, the value of the given expression in terms of i and j is -78i-12j.

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Answer:

2.

Step-by-step explanation:

To find the slope of the line that passes through the points (2,-4) and (5,2), we can use the slope formula:

slope = (y2 - y1) / (x2 - x1)

where (x1, y1) = (2,-4) and (x2, y2) = (5,2)

slope = (2 - (-4)) / (5 - 2)

slope = 6 / 3

slope = 2

Therefore, the slope of the line that passes through the points (2,-4) and (5,2) is 2.

Answer:

48 square inches

Step-by-step explanation:

lxwxh

6x4x2

The graph shown corresponds to someone who makes: C. $20 an hour.

In Mathematics and Geometry, a proportional relationship refers to a type of relationship that produces equivalent ratios and it can be modeled or represented by the following mathematical equation:

y = kx

Where:

Next, we would determine the constant of proportionality (k) by using various data points as follows:

Constant of proportionality, k = y/x

Constant of proportionality, k = 20/1 = 40/2 = 60/3 = 80/4

Constant of proportionality, k = 20.

Therefore, the required linear equation or function is given by;

y = kx

y = 20x

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

Answer:

a) Two real solutions.

b) Solutions:  y = -6, y = 12

Step-by-step explanation:

The discriminant is defined as the expression b² - 4ac, which appears under the square root sign in the quadratic formula.

The value of the discriminant determines the nature of the solutions to the quadratic equation:

[tex]\boxed{\begin{minipage}{12 cm}\underline{Discriminant}\\\\$b^2-4ac$ \quad when $ax^2+bx+c=0$\\\\when $b^2-4ac > 0 \implies$ two real solutions.\\when $b^2-4ac=0 \implies$ one real solution.\\when $b^2-4ac < 0 \implies$ no real solutions (two complex conjugate solutions).\\\end{minipage}}[/tex]

To find the number and type of solutions of the given quadratic equation, first rewrite the equation in the form ax² + bx + c = 0.

[tex]\begin{aligned}(y-3)^2-10&=71\\y^2-6y+9-10&=71\\y^2-6y-1&=71\\y^2-6y-72&=0\end{aligned}[/tex]

Compare the coefficients:

Substitute the values of a, b and c into the discriminant formula:

[tex]\begin{aligned}b^2-4ac&=(-6)^2-4(1)(-72)\\&=36-4(-72)\\&=36+288\\&=324\end{aligned}[/tex]

As the discriminant of the given equation is greater than zero, there are 2 real solutions.

To solve the equation, we can use the quadratic formula:

[tex]x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\quad\textsf{when}\;\;ax^2+bx+c=0[/tex]

As we have already calculated that the discriminant is 324, we can substitute this, along with the values of a and b, into the formula:

[tex]\implies y=\dfrac{-(-6) \pm \sqrt{324}}{2(1)}[/tex]

[tex]\implies y=\dfrac{6 \pm \sqrt{324}}{2}[/tex]

Solve for y:

[tex]\implies y=\dfrac{6 \pm 18}{2}[/tex]

[tex]\implies y=3 \pm 9[/tex]

[tex]\implies y=-6, 12[/tex]

Therefore, the solutions of the given equation are:

There are 27.558 meters of wire does she have remaining after the first day.

We have to given that;

An electrician has 42.3 meters of wire to use on a job.

And, On the first day, she uses 14.742 meters of the wire.

Hence, We get;

The remaining wire does she have remaining after the first day is,

⇒ 42.3 - 14.742

⇒ 27.558 meters

Thus, There are 27.558 meters of wire does she have remaining after the first day.

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The domain is all real numbers.

The range is {y|y ≤ 16}.

Option B is the correct answer.

We have,

The given parabola opens downwards and has a vertex at (-1,16).

So,

The parabola has a maximum value at y = 16.

Also, the y-intercept of the parabola is (0,15), which is below the vertex, and the parabola intersects the x-axis at (-5,0) and (3,0).

We can use the factored form of the equation of a parabola to find its equation in this case.

The factored form is:

y = a(x - h)² + k

where (h,k) is the vertex and "a" determines whether the parabola opens upwards or downwards.

Since the parabola opens downwards, "a" is negative.

Also, we know that the vertex is (-1,16), so we have:

y = a(x + 1)² + 16

To find "a", we can use one of the x-intercepts, say (-5,0).

Substituting these values into the equation gives:

0 = a(-5 + 1)² + 16

0 = 16a

a = 0

This indicates that the parabola is actually a line passing through (0,15) and (3,0). Therefore, the equation of the parabola is:

y = -x² - 2x + 15

The domain of this function is all real numbers because there are no restrictions on the values of x that can be plugged into the equation.

However, the range of the function is limited by the maximum value of y, which is 16.

Since the parabola opens downwards, all y values less than or equal to 16 are attainable.

Therefore, the domain is all real numbers, and the range is {y | y ≤ 16}.

Therefore,

The domain is all real numbers.

The range is {y|y ≤ 16}.

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The maximum profit that can be made is $1737.82

An equation is an expression showing the relationship between numbers and variables using mathematical operators.

The profit function is the difference between the revenue and cost of a commodity. Given that:

Revenue, R(x) = 700x - 11.3x²

Cost, C(x) = 8068 - 34.25x

Profit, P(x) = Revenue - Cost

P(x) = (700x - 11.3x²) - (8068 - 34.25x)

P(x) = 665.75x - 11.3x² - 8068

The maximum profit is at P'(x) = 0; hence:

P'(x) = 665.75 - 22.6x

665.75 - 22.6x = 0

22.6x = 665.75

x = 29.45

P(29.45) = 665.75(29.45) - 11.3(29.45)² - 8068

P(29.45) = 1737.82

The profit to be made is $1737.82

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The average rate of change for the given intervals are:

To compute the average rate of change for any given interval, determine the change between the function values at the endpoints and divide this value by the difference in independent variable values. it can be expressed as follows:

Average Rate of Change = (f(b) - f(a)) / (b - a)

The average rate of change for each point in the interval is determined form the graphs and used to solve for the average rate of change

(-5, -5) and (-4, 0):

Average Rate of Change = (0 - 0) / (-4 - (-5)) = 0 / 1 = 0

(-4, 0) and (-3, 3):

Average Rate of Change = (3 - 0) / (-3 - (-4)) = 3 / 1 = 3

(-4, 0) and (-1, 3):

Average Rate of Change = (3 - 0) / (-1 - (-4)) = 3 / 3 = 1

(-3, 0) and (-1, 0):

Average Rate of Change = (0 - 0) / (-1 - (-3)) = 0 / 2 = 0

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The binomial expansion is solved and the error in Dakota's statement is the incorrect substitution of 36g^2h^2 for the correct expression

Given data ,

Dakota made a mistake because the third term of the expansion of (2g + 3h) should have been 36g2h2. The binomial theorem asserts that the expansion of (2g + 3h) is as follows:

( x + y )ⁿ = ⁿCₐ ( x )ⁿ⁻ᵃ ( y )ᵃ

Here, x = 2g and y = 3h. Since term numbers begin at 0, since we are seeking for the third term, r = 2.

So , on simplifying the equation , we get

= nC2 * (2g)⁽ⁿ⁻²⁾ * (3h)²

= (n! / (2! * (n - 2)!)) * (2g)⁽ⁿ⁻²⁾ * (3h)²

= ((n * (n - 1)) / 2) * (2g)⁽ⁿ⁻²⁾ * (3h)²

Hence , the correct expression for the third term of the expansion of (2g + 3h) is ((n * (n - 1)) / 2) * (2g)^(n - 2) * (3h)², where n is the exponent in the binomial expansion.

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Answer:

The probability of dying from heart disease based on the given information is 1/6 or approximately 0.1667. This is because the odds in favor of dying from heart disease are 1 to 5, which means there is 1 chance of dying from heart disease for every 5 chances of not dying from heart disease. To convert odds to probability, we divide the number of chances of the event occurring by the total number of chances. So, the probability of dying from heart disease is 1/(1+5) = 1/6 or approximately 0.1667.

In the given sets, the values of A or B and B and C are:

A or B = {32, 35,36, 38, 40, 41, 44, 46, 48, 50}

B and C = {32, 48}

From the question, we are to determine the values of A or B and B and C in the given set.

From the question,

The universal set is

U = {31,32,33……,50}

and

A={35,38,41,44,46,50}

B={32,36,40,44,48}

C= {31,32,41,42,48,50}

A or B means we should find the union of A and B. That is, the elements present in A or B or both

A or B = {32, 35,36, 38, 40, 41, 44, 46, 48, 50}

B and C means we should find the intersect of B and C. That is, the elements that present both in B and C

B and C = {32, 48}

Hence,

A or B = {32, 35,36, 38, 40, 41, 44, 46, 48, 50}

B and C = {32, 48}

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The amount of fabric used to make the number cube is given as follows:

C. 294 cm².

The amount of fabric used to make the number cube is represented by the surface area of the number cube.

The surface area of a cube of side length a is given by the equation presented as follows:

S = 6a².

The side length for this problem is given as follows:

a = 7 cm.

Hence the surface area of the cube is calculated as follows:

S = 6 x 7²

S = 294 cm².

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Answer:

2.45ℓ

Step-by-step explanation:

0.33ℓ x 4 = 1.32ℓ (1ℓ 320 ml)

11.12ℓ - 1.32ℓ = 9.8ℓ (total in 4 jugs)

9.8ℓ ÷ 4 = 2.45ℓ (each jug)

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Fwam and Eva are an equal distance of 69 miles away from the stadium when the time t = 3 hours

Given data ,

The formula for the distance (F) of Fwam from the stadium is:

F(t) = 327 - 42t

Since Fwam is travelling at a 42 mph pace, his distance from the stadium is reducing at a 42 mph rate.

The formula for Eva's separation from the stadium is:

E(t) = 396 - 65t

Eva's distance from the stadium also shrinks at a pace of 65 miles per hour as she drives at a speed of 65 mph.

Set F(t) = E(t) and solve for t to get the time at which Fwam and Eva are equally far from the stadium

On simplifying the equations , we get

327 - 42t = 396 - 65t

Adding 65t to both sides:

65t + 327 - 42t = 396

23t + 327 = 396

Subtracting 327 from both sides:

23t = 396 - 327

23t = 69

Dividing both sides by 23:

t = 69 / 23

t = 3 hours

Hence , at t = 3 hours after noon, both Fwam and Eva are an equal distance of 69 miles away from the stadium.

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The complete question is attached below :

Fwam and Eva are both driving along the same highway in two different cars to a stadium in a distant city. At noon, Fwam is gar miles away from the stadium and Eva is 396 miles away from the stadium. Fwam is driving along the highway at a speed of 42 miles per hour and Eva is driving at speed of 65 miles per hour. Let F represent Fwam's distance, in miles, away from the stadium t hours after noon. Let E represent Eva's distance, in miles, away from the stadium t hours after noon. Write an equation for each situation, in terms of t, and determine how far both Fwam and Eva are from the stadium at the moment they are an equal distance from the stadium.

Beth could not join the club because the highest age which is allowed here is only 55 years

We know that;

A statistical expression obtained from a list of data that refers an abnormal gap from other values.

And, The statistical rules that instruct us to divides the data or observation values into four parts.

After analyzing the stem leaf diagram, we noticed the youngest age allowed to join the club is 10 years and the club allow highest 55 years old to join.

the plot also defines that the number of members is declined with the age. There is only person in the club who is 55 years. There are few members with the age over 40 years.

hence, Beth did not join the club because her age was above the highest age that allowed in this club.

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Answer:

7

Step-by-step explanation:

Answer:

Step-by-step explanation:

I agree it's convenience and biased.

It's the easiest way to convey a survey so it is convenient.  

it is biased because not everyone has accessible to modern technology, or that particular social media app.  Not all park users use social media.  It is biased because it is targeting a younger demographic, but excludes the youngest demographic as well, that are not allowed ot have social media.

Answer:

I agree it's convenience and biased.

It's the easiest way to convey a survey so it is convenient.  

it is biased because not everyone has accessible to modern technology, or that particular social media app.  Not all park users use social media.  It is biased because it is targeting a younger demographic, but excludes the youngest demographic as well, that are not allowed ot have social media.

Step-by-step explanation:

The calculated value of the area of the figure is 230.5 sq meters

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

The composite figure

The area is the sum of the individual areas

Using the above and the area formulas as a guide, we have the following:

Area = 5 * 5 + (6 + 5) * 4 + 1/2 * 17 * (15 + 4)

Evaluate

Area = 230.5

Hence, the area is 230.5 sq meters

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The other endpoint is (11, 1).

We have,

Midpoint = (6, 2) and endpoint = (1, 3).

and, ratio of m: n = 1 :1

Using section formula

6 = ( 1 + a) / (1+1)

6 = (1+a)/2

1 + a = 12

a = 11

and, 2 = (3 + b) / (1+1)

2 = (b + 3) /2

b +3 = 4

b = 1

Thus. the other endpoint is (11, 1).

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Area of rectangular base is 18 square centimeters.

The height (given) is 6 centimeters.

The volume of the rectangular prism is 108 cubic centimeters.

For the area of the base, multiply the length of AD times length of CD

9cm × 2cm = 18cm² remember that is square unit measure:

cm² means square centimeters.

The height is the distance between the two bases. Given as AW = 6

To get Volume, multiply the area of one base by the height

18cm² × 6cm = 108cm³  

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The sinusoidal function that rises from a minimum point at (3,-2) to a maximum point at (7, 8) is y = 5 sin(π/2(x - 3)) + 3

We are given that;

Minimum point= (3,-2)

Maximum point= (7, 8)

Now,

To construct a sinusoidal function that rises from a minimum point at (3,-2) to a maximum point at (7, 8), we can use the following steps:

Find the amplitude A. The amplitude is the distance from the midline of the function to the maximum or minimum point. The midline is the average of the maximum and minimum values, which is (8 + (-2))/2 = 3. The distance from 3 to 8 or -2 is 5, so A = 5.

Find the period P. The period is the length of one cycle of the function, or the horizontal distance between two consecutive maximum or minimum points. In this case, the period is 7 - 3 = 4. The constant B is related to the period by the formula B = 2π/P, so B = 2π/4 = π/2.

Find the horizontal shift C. The horizontal shift is the amount that the function is shifted left or right from its standard position. In this case, we want the function to have a minimum point at x = 3, so we need to shift it right by 3 units. This means that C = 3.

Find the vertical shift D. The vertical shift is the amount that the function is shifted up or down from its standard position. In this case, we want the function to have a midline at y = 3, so we need to shift it up by 3 units. This means that D = 3.

Putting it all together, we get:

y = 5 sin(π/2(x - 3)) + 3

Therefore, by the function the answer will be y = 5 sin(π/2(x - 3)) + 3.

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Using the concept of similar triangles, the length of the other two sides of triangle B are: 20 cm and 25 cm

Similar triangles are  defined as triangles that possess the same shape, but then their sizes may very well vary. This means that, if two triangles are similar, then we can say that their corresponding angles are congruent and corresponding sides are in equal proportion.

Since the shortest side of triangle B is 10cm, it means that it corresponds to the shortest side of triangle A which is 8 cm.

Thus:

Ratio of corresponding sides of B:A = 10/8

Thus:

Second side of triangle B: (10/8) = (x/16)

x = 160/8

x = 20 cm

Third side of triangle B: 10/8 = y/20

y = 200/8

y = 25 cm

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The radius of the circular mouse is 3.7 inches.

The area of the circular mouse pad 45 square inches. Therefore, the radius of the circle can be found as follows:

area of the circle = πr²

where

Therefore,

area of the circular mouse = 3.14 × r²

45 = 3.14r²

divide both sides by 3.14

r² = 45 / 3.14

r² = 14.3312101911

square root both sides

r = √14.331210911

r = 3.78565714243

r = 3.7 inches

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The expression |x³| + 5 is 5 more than the absolute value of the cube of a number.

Option B is the correct answer.

We have,

The expression |x³| represents the absolute value of the cube of a number x.

This means that no matter what value x takes (positive, negative or zero), the result of the expression will always be positive.

For example, if x = 2, then |x³| = |2³| = |8| = 8.

If x = -2, then |x³| = |-2³| = |-8| = 8.

Adding 5 to the absolute value of the cube of x means adding 5 to the positive number that resulted from |x³|.

So, if we take the examples from above, if x = 2, then the expression becomes |2³| + 5 = 8 + 5 = 13.

If x = -2, then the expression becomes |-2³| + 5 = 8 + 5 = 13.

Therefore,

The expression |x³| + 5 is 5 more than the absolute value of the cube of a number.

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