What is the area of a parallelogram with a base of 38 meters and a height of 12 meters?

Answer:

21.77 After the donation

Step-by-step explanation:

3.99 Multiplied by 6 is 23.94

So 45.71 - 23.94 = 21.77

Answer:

Mean = 51.4.

Mode = 49.

Median = 52.

Range = 6.

Step-by-step explanation:

Mean = Sum of all observations / Number of observations.

Mean = (49+49+54+55+52+49+52)/7

Mean = 360/7

Mean = 51.4 (to the nearest tenth).

Mode = The most repeated values = 49 (repeated 3 times).

Range = Largest Value - Smallest Value = 55 - 49 = 6.

Median = The central value of the data.

First, arrange the data in the ascending order: 49, 49, 49, 52, 54, 55, 55.

It can be seen that the middle value is 52. Therefore, median = 52!!!

Answer:

$520

Step-by-step explanation:

Since this is a 5% discount, you must subtract 5 from 100% which is 95%.

This will be written as 0.95.

Expression: [tex]x*0.95=494\\\\0.95x=494\\\\\frac{0.95x}{0.95} =\frac{494}{0.95}[/tex]

The answer will be $520.

Answer:

[tex]\boxed{520}[/tex]

Step-by-step explanation:

[tex]\begin{array}{rcl}\text{ Price before discount - discount} & = & \text{sale price}\\b-0.05b & = & 494\\0.95b & = & 494\\\\b & = & \dfrac{494 }{0.95}\\\\& = & \mathbf{520}\\\end{array}\\\text{The number from the set that matches } b \text{ is }\boxed{\mathbf{520}}[/tex]

Answer:

Step-by-step explanation:

It is always a good idea to start by defining variables in such a problem. Here, we can let x represent the number of calling minutes, and y represent the number of data minutes. The the total number of minutes used is ...

  x + y = 85

The total of charges is the sum of the products of charge per minute and minutes used:

  7/20x + 2/5y = 31.00

We can eliminate the x-variable in these equations by multiplying the first by -7 and the second by 20, then adding the result.

  -7(x +y) +20(7/20x +2/5y) = -7(85) +20(31)

  -7x -7y +7x +8y = -595 +620 . . . . eliminate parentheses

  y = 25 . . . . . . . . simplify

Then the value of x is

  x = 85 -y = 85 -25

  x = 60

Answer:

The second, fourth and B option are correct.

Step-by-step explanation:

In order to solve this problem, we are going to define the following variables :

[tex]X:[/tex] ''Minutes she used her pay-as-you-go phone for a call''

[tex]Y:[/tex] ''Minutes of data she used''

Then, we are going to make a linear system of equations to find the values of [tex]X[/tex] and [tex]Y[/tex].

''This month, she used a total of 85 minutes'' ⇒

[tex]X+Y=85[/tex]  (I)

(I) is the first equation of the system.

''The bill was $31'' ⇒

[tex](\frac{7}{20})X+(\frac{2}{5})Y=31[/tex] (II)

(II) is the second equation of the system.

The system of equations will be :

[tex]\left \{ {{X+Y=85} \atop {(\frac{7}{20})X+(\frac{2}{5})Y=31}} \right.[/tex]

The second option ''The system of equations is [tex]X+Y=85[/tex] and [tex](\frac{7}{20})X+(\frac{2}{5})Y=31[/tex] .'' is correct

Now, to solve the system, we can eliminate the x-variable from the equations by multiplying the equation with the fractions by 20 and multiplying the other equation by -7. Then, we can sum them to obtain the value of [tex]Y[/tex] :

[tex]X+Y=85[/tex] (I)

[tex](\frac{7}{20})X+(\frac{2}{5})Y=31[/tex] (II) ⇒

[tex](-7)X+(-7)Y=-595[/tex] (I)'

[tex]7X+8Y=620[/tex] (II)'

If we sum (I)' and (II)' ⇒

[tex](-7)X+(-7)Y+7X+8Y=-595+620[/tex] ⇒ [tex]Y=25[/tex]

If we replace this value of [tex]Y[/tex] in (I) ⇒

[tex]X+Y=85\\X+25=85\\X=60[/tex]

The fourth option ''To eliminate the x-varible from the equations, you can multiply the equation with the fractions by 20 and multiply the other equation by -7'' is correct.

With the solution of the system :

[tex]\left \{ {{X=60} \atop {Y=25}} \right.[/tex]

We answer that the option ''B-She used 60 minutes for calling and 25 minutes for data'' is correct.

Answer:

  11.28 in²

Step-by-step explanation:

The area of a Reuleaux triange is given by ...

  A = (1/2)(π -√3)d² . . . . . where d is the diameter of the triangle.

For a triangle of diameter 4 in, the area is ...

  A = (1/2)(π -√3)(4 in)² = (π -√3)8 in² ≈ 11.28 in²

_____

A Reuleaux triangle is the shape of smallest area that has a constant diameter. The diameter of the shape is the radius of each of the arcs between the vertices of the inscribed equilateral triangle.

Answer:

10 in

Step-by-step explanation:

So we want to consider the surface area of this sphere.

The formula for surface area of a sphere is [tex]A=4 \pi r^2[/tex].

So we have that the surface area is [tex]100 \pi[/tex].

So I'm going to replace [tex]A[/tex] in [tex]A=4 \pi r^2[/tex] with  [tex]100 \pi[/tex].

[tex]100\pi=4\pi r^2[/tex]

Now our main objective here is to solve for [tex]r[/tex]:

Divide both sides by [tex]4 \pi[/tex]:

[tex]\frac{100\pi}{4 \pi}=\frac{4\pi r^2}{4 \pi}[/tex]

This gets us [tex]r^2[/tex] by itself:

[tex]25=r^2[/tex]

What number squared gives you 25? If you don't know, just take the square root of both sides giving you:

[tex]\sqrt{25}=r[/tex]

Now you can just put [tex]\sqrt{25}[/tex] in your calculator.  You should get 5.

5 is the radius

The diameter is twice the radius.

So 2(5) is 10, so the diameter is 10 in.

Answer:

C) 10.0 in.

Step-by-step explanation:

If Hasan is painting a spherical model of a human cell for his science class and uses 100π square inches of paint (in one coat) to evenly cover the outside of the cell, the diameter of Hasan’s cell model is 10.0 inches.

Radius = 5

Diameter = Radius x 2

Therefore 5 x 2 = 10

The diameter is 10 in.

Answer:

  see below

Step-by-step explanation:

Each reflection is along a line through the other focus of the conic. The two foci of the parabola are the one shown and the one at infinity (the source of light rays).

The light ray in the telescope will be reflected by each conic surface in a specific manner: converging at the parabolic mirror, diverging at the hyperbolic mirror, and converging again at the elliptical mirror.

The diagram shows a telescope fitted with different types of mirror surfaces, including parabolic, hyperbolic, and elliptical mirrors. When a ray of light parallel to the parabolic axis enters the telescope, it will be reflected by each conic surface in a certain way.

The light ray will be reflected by the parabolic mirror surface and converge to a single point called the focus. This is due to the property of the parabola that all incoming parallel rays are reflected to a common focal point.

The reflected ray will then strike the hyperbolic mirror surface, where it will be reflected in such a way that it diverges outwards. Hyperbolic mirrors have a property that makes them reflect incoming parallel rays into diverging rays.

Finally, the diverging ray from the hyperbolic mirror will enter the elliptical mirror surface. The elliptical mirror will reflect the ray in such a way that it converges to a point located at the eyepiece of the telescope. Elliptical mirrors have a property that makes them reflect incoming parallel rays to a focal point.

In summary, the light ray will be reflected by the parabolic mirror surface, then the hyperbolic mirror surface, and finally, the elliptical mirror surface, converging and diverging in different ways along the way.

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Answer:

Initial height or what the ball was originally bounced from a height of 9 feet

Step-by-step explanation:

9 represents the height that the ball was originally bounced from.

If you plug in 0 for [tex]n[/tex] into [tex]f(n)=9(0.7)^n[/tex], you get:

[tex]f(0)=9(0.7)^0=9(1)=9[/tex].

9 feet is the initial height since that is what happens at time zero.

Answer:

Initial height or what the ball was originally bounced from a height of 9 feet

Step-by-step explanation:

9 represents the height that the ball was originally bounced from.

If you plug in 0 for  into , you get:

.

9 feet is the initial height since that is what happens at time zero.

Answer:

12.4 calories  / minute to the nearest tenth.

Step-by-step explanation:

That would be 560 / 45

= 12.44...  calories  / minute.

Answer:

4.97485 (approximately)

Step-by-step explanation:

You have the information SAS given.

This is a case for law of cosines.

[tex]b^2=a^2+c^2-2ac*cos(B)[/tex]

[tex]b^2=12^2+10^2-2(12)(10)*cos(24)[/tex]

[tex]b^2=144+100-240cos(24)[/tex]

[tex]b^2=244-240cos(24)[/tex]

Take the square root

[tex]b=\sqrt{244-240cos(24)}[/tex]

I was saving rounding to the end that is why I didn't put 240*cos(24) in my calculator.

So now I'm going to put sqrt(244-240*cos(24)) in my calculator. Make sure your calculator says deg (for degrees).

4.97485 (approximately)

Answer:

Circumference = 25m

Area = 50 m2

Step-by-step explanation:

formula for circumference of a circle is π(d)

when radius is 4m, diameter is 8m

3.14(8)= 25.13

nearest tenth = 25m

formula for area of circle is 2πr or π(r)(r)

when radius is 4m

3.14(4)(4)=50.27 m2

nearest tenth =50m

Answer: circumference of the circle is 25.12 m  and the area of the circle is 50.2 m²

Step-by-step explanation:

To find the circumference of the circle of radius  4 meters, we simply use the formula;

area of a circumference = 2πr

                      π is given to be 3.14 and radius r=4 meter, we will substitute this variable into the formula

area of a circumference = 2πr

                                          = 2 × 3.14 × 4

                                          =25.12

                                           ≈25.1  to the nearest tenth

Therefore, the circumference of the circle is is 25.1 meters

To find the area of the circle, we simply use the formula:

area of circle = π[tex]r^{2}[/tex]

                     =  3.14 × (4)²

                      =3.14 × 16

                        =50.24

                        ≈50.2   to the nearest tenth

Therefore, the area of the circle is 50.2 m²

Answer:

Step-by-step explanation:

The trig identities are helpful for this.

  sin² θ = 1 - cos² θ = 1 -(1/6)² = 35/36

  sin θ = (√35)/6 . . . . . . take the square root

__

  tan² θ = sec² θ -1 = (1/cos² θ) -1 = 6² -1 = 35

  tan θ = √35 . . . . . . . . . take the square root

Answer:

a_{n}=20+4(n-1)

Step-by-step explanation:

It is given that the number of corn stalks in rows of the field can be modeled by an arithmetic sequence.

The 5th row has 36 corn stalks. This means 5th term of the sequence is 36. i.e.

[tex]a_{5}=36[/tex]

The 12th row has 64 stalks. So,

[tex]a_{12}=64[/tex]

In order to write the explicit rule we need to find the first term(a1) and common difference(d) of the sequence.

The explicit rule for the arithmetic sequence is of the form:

[tex]a_{n}=a_{1}+(n-1)d[/tex]

Writing the 5th and 12th term in this way, we get:

[tex]a_{5}=a_{1}+4d[/tex]

[tex]a_{1}+4d=36[/tex]                                                     Equation 1

Similarly for 12th term, we can write:

[tex]a_{1}+11d=64[/tex]                                                     Equation 2

Subtracting Equation 1 from Equation 2, we get:

7d = 28

d = 4

Using the value of d in Equation 1, we get:

[tex]a_{1}+4(4)=36\\\\ a_{1}=20[/tex]

Thus, for the given sequence first term is 20 and common difference is 4. Using these values in the general explicit rule, we get:

[tex]a_{n}=20+4(n-1)[/tex]

Answer:

-3x+3

Step-by-step explanation:

The equation to the line shown is formed as follows.

It passes through the points (-6,0) and (-8,6)

The gradient of the line=Δy/Δx

=(y₂-y₁)/(x₂-x₁)

=(6-0)/(-8--6)

=6/-2

=-3

The line parallel to the line shown has the same gradient i.e -3

Therefore the line in question is

-3x+3.

Answer:

3 pounds of coffee

Step-by-step explanation:

First you have to find how much Jenny spent on coffee.

To find this out subtract 70 by 45.97.

So, 70 - 45.97 = $24.03

Now you have to find how many pounds of coffee she bought, so to find this out you have to divide 24.03 by 8.01.

So, 24.03 divided by 8.01 = 3 pounds of coffee.

Answer:

P(house)+P(interest)-P(both)=probability of P. Both subtracts the double counting.

0.25+0.74-P(Both)=0.89

P=0.10

P(neither) is the complement of P(either), which is OR. That is 1-0.89=0.11

If I can assume independence, which probably is not correct since the two are related, it is P(H)*P(not I)=0.25*0.26=0.065. Not I is 1-P(I)=0.26

Final answer:

The probability that house sales will increase but interest rates will not during the next 6 months is calculated using the addition rule for probabilities and is found to be 0.15 or 15%.

Explanation:

We are given three probabilities:

To find the probability that house sales will increase but interest rates will not during the next 6 months (P(House Sales Increase and Interest Rates Not Increase)), we can use the formula that relates the probability of the union of two events to the probability of each event and the probability of their intersection:

P(House Sales Increase or Interest Rates Increase) = P(House Sales Increase) + P(Interest Rates Increase) - P(House Sales Increase and Interest Rates Increase)

We rearrange the formula to solve for P(House Sales Increase and Interest Rates Not Increase):

P(House Sales Increase and Interest Rates Increase) = P(House Sales Increase) + P(Interest Rates Increase) - P(House Sales Increase or Interest Rates Increase)

Hence, the probability that interest rates will not increase when house sales increase is equal to 1 minus the probability that interest rates will increase. So:

P(House Sales Increase and Interest Rates Not Increase) = P(House Sales Increase) - P(House Sales Increase and Interest Rates Increase)

Plugging in the values we get:

P(House Sales Increase and Interest Rates Not Increase) = 0.25 - (0.25 + 0.74 - 0.89)

This simplifies to:

P(House Sales Increase and Interest Rates Not Increase) = 0.25 - 0.10 = 0.15

The probability that house sales will increase but interest rates will not during the next 6 months is 0.15 or 15%.

Answer:

Exact Form:

11/  5

Decimal Form:

2.2

Mixed Number Form:

2  1/ 5

Step-by-step explanation:

For the decimal, divide 11 by 5.

To get the Mixed number form, find out how many times 5 goes into 11, then then what is left over. Put the number left over the number that was dividing. 5 goes into 11 2 times, then 1 is left over, put the 1 over 5.

Answer:

30 ft

Step-by-step explanation:

This is a classic right triangle problem, where the length of the ladder represents the hypotenuse, where the ladder is lengthwise from the building is the base of the triangle, and what we are looking for is the height of the triangle.  Pythagorean's Theorem will help us find this length.

[tex]32^2-10^2=y^2[/tex] so

1024 - 100 = y^2 and

y = 30.4 so 30 feet

Final answer:

Using the Pythagorean theorem, it is determined that the ladder reaches approximately 30 feet up the building when one end is placed 10 feet from the wall.

Explanation:

To determine how far up the building the ladder will reach, we can use the Pythagorean theorem, which applies to right-angled triangles. The ladder forms the hypotenuse, the distance from the wall to the base of the ladder forms one leg, and the height the ladder reaches up the wall forms the other leg of the triangle.

Using the given lengths:

Ladder (hypotenuse) = 32 feet

Distance from the wall (adjacent) = 10 feet

Let height up the wall (opposite) be represented by y.

The Pythagorean theorem states that:

a^2 + b^2 = c^2

Where a and b are the legs of the triangle and c is the hypotenuse. Plugging in the values:

10^2 + y^2 = 32^2

100 + y^2 = 1024

y^2 = 1024 - 100

y^2 = 924

y = √924

y ≈ 30.4 feet

To the nearest foot, the ladder reaches approximately 30 feet up the wall.

Answer:

     -------------------------------------

-4  |  3     -10       7

Step-by-step explanation:

Take the coefficients of the numerator inside the division bar

Take the opposite of the number in the denominator

     -------------------------------------

-4  |  3     -10       7

Answer:

The correct option is B.

Step-by-step explanation:

The given expression is

[tex]\frac{3x^2-10x+7}{x+4}[/tex]

Here the numerator is

[tex]3x^2-10x+7[/tex]

So, the coefficients of numerator are 3, -10 and 7.

If the denominator of an expression is (x+c), then in synthetic division form -c is written on outside and coefficients of numerator are written under the sign of division(descending order of degree of terms).

The denominator of the expression is (x+4), so -4 is written outside the sign of division.

[tex]-4\overline{|3\quad -10\quad 7}[/tex]

Therefore the correct option is B.

Answer:

-1

Step-by-step explanation:

They already did the opposite reciprocal for you.

They have it in the form [tex]y=\frac{1}{4}x+b[/tex] now.

To find b you just enter (x,y)=(8,1).

Let's do that.

[tex]1=\frac{1}{4}(8)+b[/tex]

[tex]1=2+b[/tex]

Subtract 2 on both sides:

[tex]-1=b[/tex]

b=-1

Hey there! :)

Perp. to y = -4x + 3 ; intersects (8, 1)

Slope-intercept form is : y=mx+b where m = slope, b = y-intercept

So, our slope of the given equation is -4. However, our new slope is 1/4 because it is the negative reciprocal of -4. We have to use the negative reciprocal because our new line is perpendicular to our given one.

Now, using (8, 1) and our new slope (1/4), simply plug everything in to the point-slope form.

Point-slope : y-y1 = m(x - x1)

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

Simplify.

y - 1 = 1/4x - 2

Add 1 to both sides.

y = 1/4x - 1 ⇒ our new equation.

Therefore, the number that fits in the question mark is -1.

~Hope I helped!~

Answer:

Let Frank spends x amount in purchasing the magazines and newspapers.(though this is not used here)

MU is marginal utility where a customer can decide a particular way to allocate his income.

This allocation is done in a way, that the last dollar spent on purchasing a product will yield the same amount of extra marginal utility.

MU from the final magazine is 10 units while his MU from the final newspaper is also 10 units.

MU per dollar spent on magazines = [tex]\frac{10}{5}=2[/tex]

MU per dollar spent on newspapers = [tex]\frac{10}{2.5}=4[/tex]

We can see the MU per dollar spent on magazine is less than newspapers.

Therefore, according to the utility-maximizing rule, Frank should re-allocate spending from magazines to newspapers.

Answer:

He should investing more money on newspaper

Step-by-step explanation:

Given:

His MU from the final magazine and final newspaper is 10 utils, so we have:

magazine = $5 / 10 utils = $0.50 per util

newspaper = $2.50 / 10 utils = $0.25 per util

He should investing more money on newspaper  because twice the amount obtained from each dollar spent on newspapers than magazines as we can see above,

Hope it will find you well.

Answer:

48

Step-by-step explanation:

Answer:

There are 1,679,616 possibles outcomes

Step-by-step explanation:

This can be calculated using a rule of multiplication as:

     6    *      6   *      6    *    6       *      6    *      6     *      6     *      6     =  1,679,616

1st Roll      2nd       3rd       4th          5th        6th       7th          8th

Because the die is rolled 8 times and every roll has 6 possibilities.

Then, if the order matter, there are 1,679,616 possible outcomes.

Answer:

  84%

Step-by-step explanation:

The empirical rule tells you that 68% of the standard normal distribution is within 1 standard deviation of the mean. The distribution is symmetrical, so the amount in the lower tail is (1 -68%)/2 = 16%.

Since the number you're interested in, 240, is one standard deviation above the mean (200 +40), the percentage of interest is the sum of the area of the central part of the distribution along with the lower tail:

  68% + 16% = 84%.

Answer:

(- 1, 4 )

Step-by-step explanation:

x = 1 is a vertical line passing through all points with an x- coordinate of 1

The point P(3, 4) is to units to the right of x = 1.

Hence the refection will be 2 units to the left of x = 1

P' = (1 - 2, 4 ) = (- 1, 4 )

Answer:

[tex]\frac{2x-20}{x^2+6x-40}[/tex]

Step-by-step explanation:

[tex]f(x)+g(x)[/tex]

[tex]\frac{x-16}{x^2+6x-40}+\frac{1}{x+10}[/tex]

I'm going to factor that quadratic in the first fraction's denominator to figure out what I need to multiply top and bottom of the other fraction or this fraction so that I have a common denominator.

I want a common denominator so I can write as a single fraction.

So since the leading coefficient is 1, all we have to do is find two numbers that multiply to be c and at the same thing add up to be b.

c=-40

b=6

We need to find two numbers that multiply to be -40 and add to be 6.

These numbers are 10 and -4 since (10)(-4)=-40 and 10+-4=6.

So the factored form of [tex]x^2+6x-40[/tex] is [tex](x+10)(x-4)[/tex].

So the way the bottoms will be the same is if I multiply top and bottom of my second fraction by (x-4).

This will give me the following sum so far:

[tex]\frac{x-16}{x^2+6x-40}+\frac{x-4}{x^2+6x-40}[/tex]

Now that the bottoms are the same we just need to add the tops and then we are truly done:

[tex]\frac{(x-16)+(x-4)}{x^2+6x-40}[/tex]

[tex]\frac{x+x-16-4}{x^2+6x-40}[/tex]

[tex]\frac{2x-20}{x^2+6x-40}[/tex]

Answer:

A. (x + 1, y − 4), reflection over y = x − 4

Step-by-step explanation:

You must perform all the composed transformations to spot the one in which the coordinates of the preimage and the image are not the same.

The coordinates of the preimage are A(0,1), B(3,4), C(5,2) , and D(2,-1)

Option A is a translation (x + 1, y − 4), followed by a reflection over y = x − 4.

[tex]A(0,1)\to(1,-3)\to A'(1,-3)[/tex]

[tex]B(3,4)\to(4,0)\to B'(4,0)[/tex]

[tex]C(5,2)\to(6,-2)\to C'(2,2)[/tex]

[tex]D(2,-1)\to(3,-5)\to D'(-1,-1)[/tex]

Option B is a translation  (x − 4, y − 4), followed by a reflection over y = −x

[tex]A(0,1)\to(-4,-3)\to A'(0,1)[/tex]

[tex]B(3,4)\to(-1,0)\to B'(3,4)[/tex]

[tex]C(5,2)\to(1,-2)\to C'(5,2)[/tex]

[tex]D(2,-1)\to(-2,-5)\to D'(2,-1)[/tex]

Option C is a translation  (x +3, y − 3), followed by a reflection over y = x-4

[tex]A(0,1)\to(3,-2)\to A'(0,1)[/tex]

[tex]B(3,4)\to(6,1)\to B'(3,4)[/tex]

[tex]C(5,2)\to(8,-1)\to C'(5,2)[/tex]

[tex]D(2,-1)\to(5,-4)\to D'(2,-1)[/tex]

Option D is a translation  (x +4, y + 4), followed by a reflection over y = −x+8

[tex]A(0,1)\to(4,5)\to A'(0,1)[/tex]

[tex]B(3,4)\to(7,8)\to B'(3,4)[/tex]

[tex]C(5,2)\to(9,6)\to C'(5,2)[/tex]

[tex]D(2,-1)\to(6,3)\to D'(2,-1)[/tex]

The correct choice is A.

Answer:

A. (x + 1, y − 4), reflection over y = x − 4

Step-by-step explanation:

The answer A. (x + 1, y − 4), reflection over y = x − 4 is right because I got it right on my test!! :)))

Answer:

  (-2, -2)

Step-by-step explanation:

Compare the two functions ...

  f(x) = -|x +2| -2

  f(x) = a·g(x -h) +k

where f(x) is a translation and scaling of function g(x). Here, you have ...

  g(x) = |x|

The scale factor is a = -1.

The horizontal shift is h = -2.

The vertical shift is k = -2.

_____

The original vertex at (0, 0) has been shifted by (h, k) to ...

  (0, 0) + (h, k) = (0, 0) + (-2, -2) = (-2, -2).

Answer:

(-2,-2)

Step-by-step explanation:

I took a test and got it right

Answer:

No, the triangle is not possible.

Step-by-step explanation:

Given,

A triangle ABC in which C = 38°, a = 19, c = 10,

Where, angles are A, B and C and the sides opposite to these angles are a, b and c respectively,

By the law Sines,

[tex]\frac{sin A}{a}=\frac{sin C}{c}[/tex]

[tex]\implies sin A = \frac{a sin C}{c}[/tex]

By substituting the values,

[tex]sin A = \frac{19\times sin 38^{\circ}}{10}[/tex]

[tex]=1.16975680312[/tex]

[tex]\implies A=sin^{-1}(1.16975680312)[/tex] = undefined

Hence, the triangle is not possible with the given measurement.

Answer:

[tex](x+\frac{b}{2a})^2+(\frac{4ac}{4a^2}-\frac{b^2}{4a^2})=0[/tex]

[tex](x+\frac{b}{2a})^2=\frac{b^2-4ac}{4a^2}[/tex]

[tex]x=\frac{-b}{2a} \pm \frac{\sqrt{b^2-4ac}}{2a}[/tex]

Step-by-step explanation:

[tex]x^2+\frac{b}{a}x+\frac{c}{a}=0[/tex]

They wanted to complete the square so they took the thing in front of x and divided by 2 then squared.  Whatever you add in, you must take out.

[tex]x^2+\frac{b}{a}x+(\frac{b}{2a})^2+\frac{c}{a}-(\frac{b}{2a})^2=0[/tex]

Now we are read to write that one part (the first three terms together) as a square:

[tex](x+\frac{b}{2a})^2+\frac{c}{a}-(\frac{b}{2a})^2=0[/tex]

I don't see this but what happens if we find a common denominator for those 2 terms after the square.  (b/2a)^2=b^2/4a^2 so we need to multiply that one fraction by 4a/4a.

[tex](x+\frac{b}{2a})^2+\frac{4ac}{4a^2}-\frac{b^2}{4a^2}=0[/tex]

They put it in ( )

[tex](x+\frac{b}{2a})^2+(\frac{4ac}{4a^2}-\frac{b^2}{4a^2})=0[/tex]

I'm going to go ahead and combine those fractions now:

[tex](x+\frac{b}{2a})^2+(\frac{-b^2+4ac}{4a^2})=0[/tex]

I'm going to factor out a -1 in the second term ( the one in the second ( ) ):

[tex](x+\frac{b}{2a})^2-(\frac{b^2-4ac}{4a^2})=0[/tex]

Now I'm going to add (b^2-4ac)/(4a^2) on both sides:

[tex](x+\frac{b}{2a})^2=\frac{b^2-4ac}{4a^2}[/tex]

I'm going to square root both sides to rid of the square on the x+b/(2a) part:

[tex]x+\frac{b}{2a}=\pm \sqrt{\frac{b^2-4ac}{4a^2}}[/tex]

[tex]x+\frac{b}{2a}=\pm \frac{\sqrt{b^2-4ac}}{2a}[/tex]

Now subtract b/(2a) on both sides:

[tex]x=\frac{-b}{2a} \pm \frac{\sqrt{b^2-4ac}}{2a}[/tex]

Combine the fractions (they have the same denominator):

[tex]x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}[/tex]

bearing in mind that 4¾ is simply 4.75.

[tex]\bf ~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$600\\ r=rate\to 5\%\to \frac{5}{100}\dotfill &0.05\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\dotfill &1\\ t=years\dotfill &3 \end{cases} \\\\\\ A=600\left(1+\frac{0.05}{1}\right)^{1\cdot 3}\implies A=600(1.05)^3\implies A=694.575 \\\\[-0.35em] ~\dotfill[/tex]

[tex]\bf ~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$750\\ r=rate\to 4.75\%\to \frac{4.75}{100}\dotfill &0.0475\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\dotfill &1\\ t=years\dotfill &3 \end{cases} \\\\\\ A=750\left(1+\frac{0.0475}{1}\right)^{1\cdot 3}\implies A=750(1.0475)^3\implies A\approx 862.032[/tex]

well, the interest for each is simply A - P

695.575 - 600 = 95.575.

862.032 - 750 = 112.032.