A carpenter makes miniature replicas of Victorian furniture. The scale model of a table that he made is 3 inches long. The full-size table is 37.5 inches long. What scale was applied to the replica?

The solutions to the equation x² - 18x + 81 = 4 by completing the square are x = 9 + 2i and x = 9 - 2i.None of the answer choices A, B, C, or D are correct.

An equation is a mathematical statement that shows the equality between two expressions, often containing variables and mathematical operations.

To solve the equation x² - 18x + 81 = 4 by completing the square, we can follow these steps:

x² - 18x + 77 = 0

Divide both sides by the coefficient of x² to make the coefficient 1:

x² - 18x + (81/1) = -77/1

x² - 18x + 81 = -77

Add the square of half the coefficient of x to both sides of the equation:

x² - 18x + (9)² = -77 + (9)²

x² - 18x + 81 = -4

(x - 9)² = -4

x - 9 = ±√(-4)

x - 9 = ±2i (where i is the imaginary unit)

x = 9 ±2i

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The volume of the two cuboid added together will be 192 cm³.

what exactly is a cuboid?

A cuboid, also known as a rectangular prism, is a three-dimensional solid shape that has six rectangular faces. It is a type of polyhedron, a geometric figure with flat faces and straight edges.

A cuboid has three pairs of congruent and parallel faces, with each pair being congruent to the other. These pairs of opposite faces are known as bases, and the other four faces are called lateral faces. The lateral faces are also rectangles and are perpendicular to the bases.

Now,

As Volume of the cuboid= L*B*H

where l=length, B=Breadth and H=Height

and volume of the figure =volume of 2 cuboids

=8*4*3+10*3*4

=72+120

=192 cm³

Hence,

           The volume of the two cuboid added together will be 192 cm³.

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The perihelion is the point in a planet's orbit where it is closest to the sun.

The equation for the orbit of a planet around the sun depends on various factors, such as the shape of the orbit, the mass of the planet, and the gravitational force between the planet and the sun. Kepler's laws of planetary motion provide a framework for understanding the motion of planets in orbit, and the equations used to describe these orbits are generally based on these laws. The specific equation for the orbit of planet A would depend on the specific parameters of that planet's orbit.

Answer:

The equilibrium price is $250.

The equilibrium quantity is 26 units.

Step-by-step explanation:

To find the equilibrium price, we need to find the price at which the quantity demanded equals the quantity supplied. This occurs when q (quantity demanded) equals q (quantity supplied).

So, we can set the two equations for q equal to each other:

(380-p)/5 = (p-172)/3

To solve for p, we can cross-multiply and simplify:

3(380-p) = 5(p-172)

1140 - 3p = 5p - 860

1140 + 860 = 8p

2000 = 8p

p = 250

Therefore, the equilibrium price is $250. We can plug this value back into either equation for q to find the equilibrium quantity.

Using q = (380-p)/5, we get:

q = (380-250)/5 = 26

So the equilibrium quantity is 26 units.

This shows that any increase in x by a certain factor results in an increase in y by the same factor, confirming that y varies directly with x.

What is Linear equation ?

Linear equation can be defined as equation in which highest degree is one.

To determine if y varies directly with x, we need to check if there is a constant ratio between y and x. In other words, if we increase x by a certain factor, does y also increase by the same factor?

The equation 3y = -7x - 18 can be rewritten as y = (-7/3)x - 6. This is in the form of y = kx + b, where k is the constant of variation and represents the ratio between y and x.

Since the equation is in this form, we can say that y varies directly with x, and the constant of variation is k = -7/3.

To verify that y varies directly with x, we can check that any increase in x by a certain factor results in an increase in y by the same factor, as given by the constant of variation. For example, if we increase x by 3, then y will increase by (-7/3)(3) = -7. If we increase x by 6, then y will increase by (-7/3)(6) = -14.

Therefore, This shows that any increase in x by a certain factor results in an increase in y by the same factor, confirming that y varies directly with x.

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The equation of the line in slope-intercept form that goes through the point (8, -2) and has a slope of 1/4 is y = 1/4x - 4.

In slope-intercept form, the equation of a line is expressed as y = mx + b, where m denotes the slope of the line and b the y-intercept, or the location at where the line intersects the y-axis. In this form, the slope m denotes the line's steepness or the rate at which y changes in relation to x. A positive slope causes the line to go upward from left to right, whereas a negative slope causes the line to move downward from left to right. The value of y when x is 0, or the line's origin, is represented by the y-intercept, or b.

Given that, point (8, -2) and a slope of 1/4.

The slope-intercept form is given as:

y - y1 = m(x - x1)

Substituting the values:

y - (-2) = 1/4(x - 8)

y + 2 = 1/4x - 2

y = 1/4x - 4

Hence, the equation of the line in slope-intercept form that goes through the point (8, -2) and has a slope of 1/4 is y = 1/4x - 4.

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The slope of the line is -50 and it means that the amount of money in the account is decreasing at a rate of $50 every week.

What is meant by the slope of the line?

A line's slope is defined as the ratio of the change in the y coordinates to the change in the x coordinate. Both the net change in the y-coordinate and the net change in the x-coordinate are denoted by y and x, respectively.

As a result, m = change in y/change in x is the formula for the change in the y-coordinate with respect to the change in the x-coordinate.

where "m" represents a line's slope.

A line's slope provides information on the steepness and direction of the line. By calculating the difference between the coordinates of the two points, (x1,y1) and (x2,y2), it is simple to calculate the slope of a straight line between them.

The complete question is given below.

The two points on the graph are (4, 2400) and (12, 2000).

(x₁ , y₁) = (4, 2400)

(x₂ , y₂) = (12, 2000)

The slope of the graph can be found using the following formula.

Slope m = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex] = [tex]\frac{2000-2400}{12-4}[/tex] = [tex]\frac{-400}{8}[/tex] = -50

Therefore the slope of the line is -50 and it means that the amount of money in the account is decreasing at a rate of $50 every week.

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Therefore , the solution of the given problem of coordinates comes out to be , T's values are (15, 7).

A coordinate system can be used to precisely find points or additional mathematical objects on such a space, including Euclidean space, by using one or more variables or coordinates. To find a point or item on a double plane, one uses coordinates, which are pairs of numbers. Two numbers called the y and x matrices are used to describe a point's location on a two-dimensional plane. a set of numbers used to identify specific locations. The number usually consists of two digits.

Here,

In other terms, the x-coordinate of T is 6 times the difference between t and 6, or:

=> T's x-coordinate is 6 plus (6 minus (-3)) = 15

We can use the fact that the line of reflection is just the perpendicular bisector of the section joining t and to determine the y-coordinate of T. T's separation from the line

=>  x=6 is 6 - (-3) = 9,

which is also T's separation from the line x=6.

The y-coordinate of T is the same as the y-coordinate of t because the line of reflection is the perpendicular bisector of the section joining t and T, which is:

=>T has a y-coordinate of 7.

Consequently, T's values are (15, 7).

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The area of the shaded sector is equal to: A. 415π/2 ft².

Mathematically, the area of a sector can be calculated by using this formula:

Area of sector = θπr²/360

Where:

Substituting the given parameters into the area of a sector formula, we have the following;

Area of sector = 332(π/180) × (15)²/2

Area of sector = 74,700π/180 × 1/2

Area of sector = 415π/2 ft²

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Answer:

[tex]x-1/x^2 +5x+4[/tex] = no real value

Step-by-step explanation:

Step 1 : Multiply through by x^2

[tex]x^{2} -1 +5x^{3}+ 4x^{2}[/tex]

Step 2 : Collect like terms

[tex]5x^{3}+5x^{2} -1\geq 0[/tex]

Step 3 : Ignore the sign and solve for x

[tex]5x^{3} +5x^{2} -1 = 0\\x =0.380609458\\x= 0.4(2dp)[/tex]

Step 4 : Input back into the equation

Step 5: This shows that  x-1/x^2 +5x+4 has no real values.

The rock will be at a height of 125 feet after 0.49 and 4.76 seconds.

In the given problem we have a function h(t) that represents the height of the rock that is from the ground of the beach where the variable represents the time travel by the rock.

Assume t as required time equates the given function to the given height and solve for the value of 't'.

Here we have

A rock is thrown directly upward with an initial velocity of 79 feet per second from a cliff 50 feet above a beach.

The height of the rock above the beach (h) after t seconds is given by the equation h(t) = -16t² + 79t + 50.  

Let after t seconds the height will be 125 feet

=> h(t) = 125  

=>  -16t² + 79t + 50 = 125

=> -16t² + 79t - 75 = 0

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

=> x = [-b ± √(b² - 4ac)]/ 2a

Here a = -16, b = 79, and c = -75.  

t = [-79 ± √(79² - 4(-16)(-75))] / 2(-16)

t = (-79 ± √(6241 - 4800)) / -32

t = (-79 ± √1441) / -32

So, the two solutions are:

t = (-79 + √1441) / -32 and   t = (-79 - √1441) / -32

t ≈ 0.497 or t ≈ 4.763

Therefore,

The rock will be at a height of 125 feet after 0.49 and 4.76 seconds  

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After simplifying a mixed number is 5 1/16

A mixed number is a combination of a whole number and a fraction.

It is typically written in the form "a b/c", where "a" is the whole number, "b" is the numerator of the fraction, and "c" is the denominator of the fraction.

For example, 3 1/2 is a mixed number, where 3 is the whole number, 1 is the numerator of the fraction, and 2 is the denominator of the fraction. This mixed number can also be expressed as an improper fraction as follows:

3 1/2 = (3 × 2 + 1) / 2 = 7/2.

Conversely, an improper fraction can be converted to a mixed number by dividing the numerator by the denominator to obtain the whole number and expressing the remainder as a fraction.

To multiply 2 1/4, follow these steps:
1. Convert the mixed number to an improper fraction: 2 1/4 = (2 × 4 + 1) / 4 = 9/4
2. Multiply the improper fraction by itself: (9/4) × (9/4)
3. Multiply the numerators: 9 × 9 = 81
4. Multiply the denominators: 4 × 4 = 16
5. Write the result as a fraction: 81/16
6. Simplify the fraction by converting it to a mixed number:

81 ÷ 16 = 5, with a remainder of 1.

So, 81/16 = 5 1/16.

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Answer:

Step-by-step explanation:

To write the equation of the line that passes through the points (3, -4) and (7, 6), we can follow these steps:

Step 1: Find the slope of the line

The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:

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

Plugging in the given values, we get:

slope = (6 - (-4)) / (7 - 3)

slope = 10 / 4

slope = 5 / 2

Step 2: Use point-slope form to write the equation of the line

Point-slope form of a line with slope m passing through a point (x1, y1) is given by:

y - y1 = m(x - x1)

We can use either of the given points to write the equation. Let's use (3, -4):

y - (-4) = (5/2)(x - 3)

Simplifying this equation, we get:

y + 4 = (5/2)x - (15/2)

Subtracting 4 from both sides, we get:

y = (5/2)x - (23/2)

This is the equation of the line in point-slope form.

Step 3: Simplify the equation if it is not in point-slope form

The equation we obtained in step 2 is already in point-slope form, so we do not need to simplify it any further.

Note: If the line was horizontal (i.e., it had zero slope), then its equation would be y = constant, where the constant is the y-coordinate of any point on the line. If the line was vertical (i.e., its slope was undefined), then its equation would be x = constant, where the constant is the x-coordinate of any point on the line.

Answer:

The height of the patio heater is approximately 8.9 feet.

Answer:

Y=8

Step-by-step explanation:

7(y-5)=21

distribute the 7

7y-35=21

get y by itself

7y=56

divide by 7

y=8

HOPE THIS HELPS YOU UNDERSTAND!

Answer:

The correct answer is g(x) = (x+4)2+3

Step-by-step explanation:

The graph is sifted 3 units up and 4 units left.

Based on division operation, the number of boxes of papers that the back of Nico's truck can stack in is 224.

Division operation is one of the four basic mathematical operations, including addition, subtraction, and multiplication.

Division operation involves the dividend divided by the divisor to produce a result known as the quotient.

In this situation, the volume of Nico's truck is determined as the dividend. The volume of each box of paper is determined as the divisor.

The result of the division operation is the quotient showing the number of boxes the truck can contain.

The length of Nico's truck = 7 feet

The width of Nico's truck = 4 feet

The height of Nico's truck = 8 feet

The volume that Nico's tuck can contain = 224 cubic feet (7 × 4 × 8)

= 224 feet³

The length of each box = 1 foot

The width of each box = 1 foot

The height of each box = 1 foot

The volume of each box of papers = 1 feet³ (1 × 1 × 1)

The number of boxes the truck can contain = 224 boxes (224/1)

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Answer:

Let's call the height of the antenna "h".

First, we can use the angle of elevation of 38^{\circ} ∘ to find the height of point A above the ground.

tan(38^{\circ}) = \frac{h}{21}

h = 21 \cdot tan(38^{\circ})

h \approx 15.6

So point A is approximately 15.6 meters above the ground.

Next, we can use the angle of elevation of 46^{\circ} ∘ to find the height of point B above the ground.

tan(46^{\circ}) = \frac{h}{d}

h = d \cdot tan(46^{\circ})

We can find the value of "d" using the Pythagorean theorem.

d^2 = 21^2 + 15.6^2

d \approx 25.7

So the distance from point A to point B is approximately 25.7 meters.

Finally, we can use the height of point A and the distance from point A to point B to find the height of point B (the height of the antenna).

h = d \cdot tan(46^{\circ})

h \approx 25.7 \cdot tan(46^{\circ})

h \approx 23.2

Therefore, the height of the antenna is approximately 23.2 meters.

Step-by-step explanation:

the scenario creates 2 right-angled triangles.

both have the same first leg : the horizontal distance from Avani's eyes to the building (21 m).

and both have a right angle (90°) at the point, where the horizontal distance meets the building.

the difference is now the second leg : the height of the building (starting at 1.66 m above ground), and the height of the building plus the height of the antenna (again starting at 1.66 m above ground).

another difference is the length of the line of sight (from Avani to AA, and from Avani to BB).

driving these differences is the difference in the angle at Avani (38° vs. 46°).

now, remember the law of sine :

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

a, b, c are the sides of the triangle, A, B, C are the corresponding opposite angles of the triangle.

and remember : the sum of all angles in a triangle is always 180°.

what is the plan ?

we need to calculate the second leg of the larger triangle, and then the second leg of the smaller triangle and subtract that from the second leg of the larger triangle.

in other words :

(building + antenna) - building = antenna

so, we start with the larger triangle (up to BB).

the angle at Avani is 46°.

the angle at the building is 90°.

the angle at BB is then

180 - 90 - 46 = 44°.

21/sin(44) = (building + antenna)/sin(46)

(building + antenna) = 21×sin(46)/sin(44) =

= 21.74613659... m

now, for the smaller triangle (up to AA).

the angle at Avani is 38°.

the angle at the building is 90°.

the angle at AA is then

180 - 90 - 38 = 52°.

21/sin(52) = building/sin(38)

building = 21×sin(38)/sin(52) = 16.40699816... m

the height of the antenna is then again

(building + antenna) - building = 5.339138433... m

≈ 5.3 m

Answer:

The gradient is given as m=1, and the point (7,2) lies on the line. Thus:

y - y1 = m(x - x1)

y - 2 = 1(x - 7)

y - 2 = x - 7

y = x - 5

So the equation of the line is y = x - 5.

Again, using the point-slope form of a straight line:

The gradient is given as m=-2, and the point (6,-3) lies on the line. Thus:

y - y1 = m(x - x1)

y - (-3) = -2(x - 6)

y + 3 = -2x + 12

y = -2x + 9

So the equation of the line is y = -2x + 9.

The value of the expression is between 7/9 and 8, since 7/9 < 56/9 < 8. So the correct option is D.

Algebraic expressions can represent real-world situations, formulas, and equations. They are commonly used in algebra, which is a branch of mathematics that deals with symbols and the rules for manipulating these symbols.

Algebraic expressions are important tools in solving equations and real-world problems that involve variables and unknowns. They are also used in calculus, physics, engineering, and other fields that require mathematical modeling and analysis.

The value of the expression 8×7/9 can be simplified using the order of operations (PEMDAS) as follows:

8×7/9 = (8×7)/9 = 56/9

Therefore, the value of the expression is between 7/9 and 8, since:

7/9 < 56/9 < 8

So the correct statement is: D) between 7/9 and 8.

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so hmmm 25% of that 300,000 is going as a downpayment, so he need the loan for the remaining 75% of that hmmm

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{\textit{\LARGE a}\% of \textit{\LARGE b}}\\ \cline{1-1} \\ \left( \cfrac{\textit{\LARGE a}}{100} \right)\cdot \textit{\LARGE b} \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{75\% of 300000}}{\left( \cfrac{75}{100} \right)300000}\implies 225000[/tex]

so for that much, so since he'll be making monthly payments, the compounding period will be monthly, now, we're assuming the payments are at the end of each month.

[tex]~~~~~~~~~~~~\underset{\textit{payments at the end of the period}}{\textit{Payments of an ordinary annuity}} \\\\ pmt=A\left[ \cfrac{\frac{r}{n}}{\left( 1+\frac{r}{n} \right)^{nt}-1} \right][/tex]

[tex]\begin{cases} A=\textit{accumulated amount}\dotfill &225000\\ pmt=\textit{periodic payments}\\ r=rate\to 2\%\to \frac{2}{100}\dotfill &0.02\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{monthly, thus twelve} \end{array}\dotfill &12\\ t=years\dotfill &25 \end{cases}[/tex]

[tex]pmt=225000\left[ \cfrac{\frac{0.02}{12}}{\left( 1+\frac{0.02}{12} \right)^{12 \cdot 25}-1} \right] \\\\\\ pmt=225000\left[ \cfrac{\frac{1}{600}}{\left( \frac{601}{600} \right)^{300}-1} \right]\implies pmt\approx 578.67[/tex]

FG and KH are congruent using the AAS theorem. ∠F = ∠K and ∠G = ∠H

The AAS (Angle-Angle-Side) Congruence Theorem is a geometric theorem that states that if two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the two triangles are congruent.

In other words, if two triangles have two corresponding angles that are congruent, and the included side between these angles is also congruent in both triangles, then the two triangles are congruent.

The diagram shows that we have two triangles here. The first triangle is equal to the second triangle.

This is shown by the fact that the angle at F = angle at K

the angle at H = angle at G

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Answer:

use AAS for this

Step-by-step explanation:

The total of the first four terms is thus 8/3.

The outcome of adding two or even more numbers or phrases is known as the sum in mathematics. The sum is a method of bringing things together as a result. To put it another way, adding any number of numbers together results in a previous result or total.

The sum of a Fibonacci sequence that has the initial component a = 1 and the general ratio r = 5/3 is determined using the formula you gave.

We enter these numbers into the formula to determine the sum of the initial n terms:

S(n) = [tex]a(1-r^n)/(1-r)[/tex]

S(n) = [tex]1(1-(5/3)^n)/(1-5/3)[/tex]

S(n) = [tex]3/2(1-(5/3)^n)[/tex]

So, we add n to this formula to get the sum of the initial n terms of a given sequence:

S(n) = [tex]3/2(1-(5/3)^n)[/tex]

For instance, we replace n = 4 to get the sum of the initial four terms:

S(4) = [tex]3/2(1-(5/3)^4)[/tex]

S(4) = 3/2(1-625/81)

S(4) = 3/2(144/81)

S(4) = 72/27

S(4) = 8/3

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The total of the first four terms is thus 8/3.

The outcome of adding two or even more numbers or phrases is known as the sum in mathematics. The sum is a method of bringing things together as a result. To put it another way, adding any number of numbers together results in a previous result or total.

The sum of a Fibonacci sequence that has the initial component a = 1 and the general ratio r = 5/3 is determined using the formula you gave.

We enter these numbers into the formula to determine the sum of the initial n terms:

S(n) = [tex]a(1-r^{n} )/(1-r)[/tex]

S(n) = [tex]1(1-(5/3)^{n} /(1-5/3)[/tex]

S(n) = [tex]3/2(1-(5/3)^{n} )[/tex]

So, we add n to this formula to get the sum of the initial n terms of a given sequence:

S(n) = 3/2(1 - [tex](5/3)^{n}[/tex] )

For instance, we replace n = 4 to get the sum of the initial four terms:

S(4) = [tex]3/2(1-(5/3)^{4} )[/tex]

S(4) = 3/2(1-625/81)

S(4) = 3/2(144/81)

S(4) = 72/27

S(4) = 8/3

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complete ques is:

Find the sum of the first n terms ( 1,2,3,4,5.......n)

using the formula:

a(1-r^n)/1-r

1, 5/3, 25/9, 125/27, 625/81

Answer:

A) AAA yessss

Answer:

s = -1.5

Step-by-step explanation:

0.5s + 1 = 7 + 4.5s

So we can combine like terms

Put 0.5s to other side

Put 7 to other side

Then you get the equation:

1 - 7 = 4.5s - 0.5s

So we simplify:

-6 = 4s

That means

s = -6/4

s = -1.5

Hope this helps!

Answer:  What an entrepreneur really is, though, is someone who runs their own business and takes a risk to do it.

Step-by-step explanation:

Answer: To find the points where the circle intersects the line x=1, we substitute x=1 in the equation of the circle:

2(1)² + 2y² - 5(1) + 7y - 36 = 0

Simplifying, we get:

2y² + 7y - 31 = 0

We can solve this quadratic equation by using the quadratic formula:

y = (-7 ± √(7² - 4(2)(-31))) / (2(2))

y = (-7 ± √225) / 4

y = (-7 ± 15) / 4

So the two possible values of y are:

y = 2 or y = -8/2 = -4

Therefore, the points where the circle intersects the line x=1 are (1, 2) and (1, -4).

Step-by-step explanation:

Answer: To find the perimeter of the door on a scale drawing, we need to first determine the dimensions of the door on the scale drawing.

If the actual height of the door is 7 centimeters, then the height on the scale drawing will be:

7 cm ÷ 3.5 = 2 cm

Similarly, if the actual width of the door is 3 centimeters, then the width on the scale drawing will be:

3 cm ÷ 3.5 = 0.857 cm

Now we can use these dimensions to find the perimeter on the scale drawing:

Perimeter = 2 × (height + width) = 2 × (2 cm + 0.857 cm) = 2 × 2.857 cm = 5.714 cm

Therefore, the perimeter of the door on the scale drawing is 5.714 centimeters.

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