7. The top of a tree is 35 feet above ground level. The roots reach down 22 feet below ground
level. What is the difference between the top of the tree and the bottom of the roots?

The answer is x=2.

\((6^{x}) ^{10} =(6^{12} )^{2} /6^{4}\)

\(6^{10x} =(6^{12}) ^{2} -6^{4}\)  

10x=24-4

10x=20

x=20/10

x=2

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The $1314^\text{th}$ digit past the decimal point is 2.

To find the $1314^\text{th}$ digit past the decimal point in the decimal expansion of $\frac{5}{14}$, we can use long division to compute the decimal expansion of the fraction.

The long division of $\frac{5}{14}$ is as follows:

```

0.35    <-- Quotient

-----

14 | 5.00

   4.2    <-- Subtract: 5 - (14 * 0.3)

   -----

     80   <-- Bring down the 0

     70    <-- Subtract: 80 - (14 * 5)

     -----

     100   <-- Bring down the 0

      98    <-- Subtract: 100 - (14 * 7)

      -----

       20   <-- Bring down the 0

       14    <-- Subtract: 20 - (14 * 1)

       -----

        60   <-- Bring down the 0

        56    <-- Subtract: 60 - (14 * 4)

        -----

         40   <-- Bring down the 0

         28    <-- Subtract: 40 - (14 * 2)

         -----

         120   <-- Bring down the 0

         112    <-- Subtract: 120 - (14 * 8)

         -----

           80   <-- Bring down the 0

           70    <-- Subtract: 80 - (14 * 5)

           -----

           ...

```

We can see that the decimal expansion of $\frac{5}{14}$ is a repeating decimal pattern with a repeating block of digits 285714. Therefore, the $1314^\text{th}$ digit past the decimal point is the same as the $1314 \mod 6 = 0^\text{th}$ digit in the repeating block.

Since $1314 \mod 6 = 0$, the $1314^\text{th}$ digit past the decimal point in the decimal expansion of $\frac{5}{14}$ is the first digit of the repeating block, which is 2.

So, the $1314^\text{th}$ digit past the decimal point is 2.

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Answer:

I've attached the graph with it, the intersection point is,

x = 3, y = 2

Answer:

Do the grouping 3x² - 3xy - 2x + 2y =

( 3x² - 3xy ) + ( -2 + 2y ) and factor out 3x in the first and -2 in the second group.

3x ( x-y ) -2 ( x-y)

Factor out common term x-y by using distributive property.

So the answer is ( x - y ) ( 3x - 2 )

Hope it's Help

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The other side lengths of the triangle are 26 and 28 cm

The perimeter (P) is given as:

P = 84 cm

The area is given as:

A = 336

Let the sides be x, y and z.

So, we have:

x + y + z = 84

By herons' formula, we have:

\(Area = \sqrt{s(s-x)(s-y)(s-z)}\)

Where:

s = 0.5(x + y + z)

Multiply by 2

2s = x + y + z

Recall that:

x + y + z = 84

So, we have:

2s = 84

Divide by 2

s = 42

Let x = 30 ---- the given side length

So, we have:

30 + y + z = 84

Subtract 30 from both sides

y + z = 54

Make y the subject

y = 54 - z

Recall that

\(Area = \sqrt{s(s-x)(s-y)(s-z)}\)

The area is 336. So, we have:

\(336 = \sqrt{s(s-x)(s-y)(s-z)}\)

Square both sides

112896 = s(s-x)(s-y)(s-z)

Substitute values for x, y and s

112896 = 42(42-30)(42 - (54 - z))(42 - z)

Divide through by 42

2688 = (42-30)(42 - (54 - z))(42 - z)

Divide through by 12

224 = (42 - (54 - z))(42 - z)

Evaluate the brackets

224 = (z - 12)(42 - z)

Using a graphing calculator, we have:

z =26 or z = 28

Recall that:

y = 54 - z

So, we have

y = 54 - 26 = 28

y = 54 - 28 = 26

Hence, the other side lengths of the triangle are 26 and 28 cm

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=======================================================

Explanation:

Triangle ABC is similar to triangle DEF. The order of the lettering is important because it tells us how the letters pair up.

Based on that, we can determine how the segments pair up

We want to find how long EF is, which means we'll involve BC = 17 since it is paired with it.

The diagram shows that AC = 14 and DF = 3, which is another pair we'll use

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

The key takeaway is that

Those two facts lead to the steps below

\(\frac{AC}{DF} = \frac{BC}{EF}\\\\\frac{14}{3} = \frac{17}{EF}\\\\14*EF = 3*17\\\\14*EF = 51\\\\EF = \frac{51}{14}\\\\EF \approx 3.642857\\\\EF \approx 3.6\\\\\)

The proportion in step 1 is valid due to the similar triangles, and because of the corresponding side pairs mentioned. In step 3, I cross multiplied.

Answer:

3.6 cm

Step-by-step explanation:

Similar Triangle Theorem

If two triangles are similar, the ratio of their corresponding sides is equal.

Given triangles:

Therefore the corresponding sides are:

To find the measure of side EF, use the Similar Triangle Theorem:

\(\implies \sf AC:DF=BC:EF\)

\(\implies \sf \dfrac{AC}{DF}=\dfrac{BC}{EF}\)

\(\implies \sf \dfrac{14}{3}=\dfrac{17}{EF}\)

\(\implies \sf EF=17 \cdot \dfrac{3}{14}\)

\(\implies \sf EF=\dfrac{51}{14}\)

\(\implies \sf EF=3.642857...\)

\(\implies \sf EF=3.6\:cm\:\:(nearest\:tenth)\)

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Answer:

uhhh... why dont you check my work lol

Step-by-step explanation:

y = -15/18x + b

-1 = -15/18 (6) + b

-1 = 5 + b

-6 = b

check your work:

14 = -15/18(-12) - 6

14 = 10 - 6

14 = 4

The donuts are cheaper in bakery B which implies that Kevin is right.

In order to solve a problem for two different values of a quantity, its unit value is first derived. This method is known as unitary method. The ratio of two numbers are taken in this method. There can be two relations between the ratio a direct or n indirect. For the direct relation the quantities simultaneously increase or decrease while in the indirect it happens reverse.

The price of donuts for both bakeries are given as,

For Bakery A,

The price of 10 donuts is $15.

Unit rate is given as 15/10 = $1.50 per donut.

For Bakery B,

The price of 12 donuts is $17.

Unit rate is given as 17/12 = $1.41 per donut.

Thus, price of donuts from Bakery B is less than Bakery A.

It implies that Kevin is right.

Hence, the price of donuts is cheaper in case of bakery B which implies that Kevin is right.

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Kevin is correct. And he used the correct unit rate formula to find the inequality.

In mathematics, "inequality" refers to a relationship between two expressions or values that are not equal to each other. To solve the inequality, you may multiply or divide each side by the same positive number, add the same amount to each side, take the same amount away from each side, and more. You must flip the inequality sign if you multiply or divide either side by a negative number.

Given:

Bakery A.

She used the equation 10 donuts $15 to show that their donuts cost about $0.67 per donut.

This is less than Bakery B's price, 12 donuts $17= about $0.71 per donut.

Kevin thinks they should order donuts from Bakery B.

He used the equation

$17 ÷ 12 donuts to show that their donuts cost about $1.42 per donut.

This is less than Bakery A's price,

$15÷ 10 donuts = about $1.50 per donut.

The inequality:

1.42 < 1.50

To find the unit rate:

Unit rate = the total price of donut / the number of total donuts

Kevin used the above formula to find the unit rate.

So, Kevin is right.

Therefore, Kevin is right.

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Answer:

x>9 and x<11

Step-by-step explanation:

The length of the shadow casted by the high rise building is approximately 37.7 feet

The image in the question forms a right triangle:

Angle θ = 57 degrees

Opposite to angle θ = 58 feet

Adjacent to angle θ = x

To solve for x ( length of the shadow casted by the building ), we use the trigonometric ratio.

Note: tangent = opposite / adjacent

Hence:

tan( θ ) = opposite / adjacent

Plug in the values:

tan( 57° ) = 58ft / x

Cross multiply and solve for x:

x × tan( 57° ) = 58ft

x = 58ft / tan( 57° )

x = 37.7 ft

Therefore, the value of x is 37.7 feet.

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Answer:

5. a) y = 4

6. c) x = 2

Step-by-step explanation:

Answer:

y = 4 and x = 2

Step-by-step explanation:

According to the linear equation, he has to sold 50 products to collect a amount of 5000 rs.

According to the statement

We have to find that the value of the products he sold.

So, For this purpose, we know that the

The definition of a linear equation is an algebraic equation in which each term has an exponent of one and the graphing of the equation results in a straight line.

From the given information:

Chip earns a base salary of $500 per month as a salesman. In addition to the salary, he earns $90 per product that he sells. If his goal is to earn $5000 per month.

Then

The equation become

90x + 500 = 5000

Here x is the product he sold

Then

90x + 500 = 5000

90x  = 5000 - 500

90x = 4500

x = 4500/90

x =  50.

So, According to the linear equation, he has to sold 50 products to collect a amount of 5000 rs.

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Answer:

It is 9

Step-by-step explanation:

\(4(y + z)\)

when y is 5, z is 4

substitute:

\( = 4(5 + 4) \\ = 36\)

Answer:

y=5 z = 4 - substitution

4 (5 + 4) = 4 (9) = 36

Shaquille has the highest typing speed of 45 WPM, making him the fastest typist among the four individuals. Therefore, the answer is d).

To determine the fastest typist among Ella, Harper, Owen, and Shaquille, we need to calculate their typing speeds in words per minute (WPM). Typing speed is calculated by dividing the number of words typed by the minutes spent typing.

For Ella, she typed 640 words in 16 minutes, resulting in a typing speed of 640/16 = 40 WPM.

For Harper, she typed 450 words in 15 minutes, resulting in a typing speed of 450/15 = 30 WPM.

For Owen, he typed 560 words in 14 minutes, resulting in a typing speed of 560/14 = 40 WPM.

For Shaquille, he typed 540 words in 12 minutes, resulting in a typing speed of 540/12 = 45 WPM.

Comparing the typing speeds, we find that Shaquille has the highest typing speed of 45 WPM, making him the fastest typist among the four individuals.

Therefore, the answer is d) Shaquille.

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According to the information in the table, who is the fastest typist?

Words Typed by Four Typists

Typist         Words Typed       Minutes Typing

Ella                   640                             16

Harper            450                             15

Owen               560                              14

Shaquille        540                             12

a) Ella

b) Harper

c) Owen

d) Shaquille

Based on the data in the table, Ella is the fastest typist because she has typed 640 words in 16 minutes.

According to the table given, Ella is the fastest typist with a typing speed of 40 words per minute.

She typed a total of 640 words in 16 minutes which is twice the number of words typed by Harper, who is the second fastest typist.

Harper has a typing speed of 30 words per minute and typed 450 words in 15 minutes.

Owen and Shaquille have typing speeds of 28 and 45 words per minute, respectively.

Owen typed 560 words in 14 minutes, whereas Shaquille typed 540 words in 12 minutes.

Therefore, Ella is the fastest typist among all the typists mentioned in the table and Shaquille is the fastest among Owen and Shaquille.

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It is given that the point G lies on the line segment FH and divides it in the ration 5:2.

So we will use the section formula to obtain the coordinates of the point G.

Now the x-coordinate of the point G will be:-

Now the y-ccordinate of the point G will be:-

So the coordinates of the point G are (-6, 14).

Hence the correct option is (D).

Answer: 128cm^2

Step-by-step explanation: Hope it helped

Answer:

a) 72

b) 1/72

c) 1/36

Step-by-step explanation:

a) number of ways she can choose route= 3C1 = 3

   number of ways she can choose parking lots= 4C1 = 4

number of ways she can choose entrances= 3C1 = 3  

number of ways she can choose elevators= 2C1 = 2

number of ways she can go to office= number of ways she can choose route×number of ways she can choose parking lots×number of ways she can choose entrances×number of ways she can choose elevators

      number of ways she can go to office= 3×4×3×2

                                                                      = 72

b) Probability of morning side= number of morning side/ total number of routes=  1/3

probabiltiy of Parking lot A=  number of parking lot A/ total number of parking lot= 1/4

probability of south entrance= number of south entrance/ total number of entrances=  1/3

probablity of elevator 1=  number of elevator 1/ total number of elevator= 1/2

combine probability= 1/3× 1/4×1/3×1/2 = 1/72

c) Probability of Industrial Avenue= number of industrial avenue/ total number of avenue= 1/3

Probability of parking lot D= number of parking lot D/ total number of parking lot after deducting number of parking lots A and B = 1/2

Probability of north entrance= number of north entrance/ total number of entrance= 1/3

probablity of elevator 2= number of elevator 2/ total number of elevator

                                      = 1/2

combine probability= 1/3 × 1/2 × 1/3 ×1/2

                                  = 1/36

Answer:

so u put the first point at -1 go down 2 to the right 1

The graph of equation y = - 2x - 1 and the y-intercept (0, - 1) is shown in image.

We have to given that,

To plot the equation y = - 2x - 1 and the y-intercept on the graph.

Since, The equation of line in slope intercept form is,

y = mx + c

Where, m is slope and c in y - intercept.

Here, Equation is,

y = - 2x - 1

After comparing,

y - intercept = - 1

So, y - intercept of equation (0, - 1)

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The number of ways in which we can selected these 5 officers are 157,872.

This is a question of permutations and combinations.

Given that:-

Number of members in the club = 24

Number of pledges in the club = 13

We have to find the number of ways in which we can selected these 5 officers.

We know that:-

Number of ways of selecting a president, vice-president, and secretary from the members = \(^{24}C_3=2024\)

Number of ways of selecting a pledge chairman and pledge vice chairman from the pledges = \(^{13}C_2=78\)

Hence, number of ways of selecting 5 officers are 2024*78 = 157,872.

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The values of x and y on the triangle are given as follows:

The three trigonometric ratios are the sine, the cosine and the tangent, and they are obtained according to the rules presented as follows:

The side x is adjacent to the angle of 30º, while the opposite side to the angle of 30º is of 5 units, hence:

tan(30º) = x/5

\(\frac{\sqrt{3}}{3} = \frac{x}{5}\)

\(x = \frac{5\sqrt{3}}{3}\)

Considering that side 5 is opposite to the angle of 30º, the hypotenuse y is obtained as follows:

sin(30º) = 5/y

1/2 = 5/y

y = 5 x 2

y = 10.

The triangle is given by the image presented at the end of the answer.

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The estimated median price of a single-family home in Charleston/No. Charleston in the 1st Quarter of 2008 is $201,048. If the median price continues to decrease at the same rate for the next two years, the estimated median price of a single-family home in the 1st Quarter of 2011 would be $144,458.

(a) To estimate the median price of a single-family home in the 1st Quarter of 2008, we need to calculate the original price before the 8.15% decrease. Let's assume the original price was P. The price after the decrease can be calculated as P - 8.15% of P, which translates to P - (0.0815 * P) = P(1 - 0.0815). Given that the end of 1st Quarter of 2009 median price was $184,990, we can set up the equation as $184,990 = P(1 - 0.0815) and solve for P. This gives us P ≈ $201,048 as the estimated median price of a single-family home in the 1st Quarter of 2008.

(b) If the median price of a single-family home falls at the same rate for the next two years, we can calculate the price for the 1st Quarter of 2011 using the estimated median price from the 1st Quarter of 2009. Starting with the median price of $184,990, we need to apply an 8.15% decrease for two consecutive years. After the first year, the price would be $184,990 - (0.0815 * $184,990) = $169,805.95. Applying the same percentage decrease for the second year, the price would be $169,805.95 - (0.0815 * $169,805.95) = $156,012.32. Therefore, the estimated median price of a single-family home in the 1st Quarter of 2011 would be approximately $144,458.

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Answer:

\(\frac{9\pm\sqrt{61}}{2}\)

Step-by-step explanation:

\(\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}=\frac{-(-9)\pm\sqrt{(-9)^2-4(1)(5)}}{2(1)}=\frac{9\pm\sqrt{81-20}}{2}\\\\=\frac{9\pm\sqrt{61}}{2}\)

The magnitude of the resultant force can be found using the law of cosines, and it is approximately 692 pounds.

The direction of the resultant force can be found using the law of sines, and it is approximately 14.6° with respect to the positive x-axis

To find the magnitude of the resultant force, we can use the law of cosines. The law of cosines states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of their magnitudes and the cosine of the included angle.

In this case, the two sides are the magnitudes of the given forces (300 pounds and 500 pounds), and the included angle is the angle between the forces.

Applying the law of cosines, we have: Resultant force^2 = 300^2 + 500^2 - 2 * 300 * 500 * cos(60° - (-45°))

Calculating this equation, we find that the resultant force^2 is approximately equal to 479,200 pounds^2. Taking the square root of this value, we get the magnitude of the resultant force, which is approximately 692 pounds.

To find the direction of the resultant force, we can use the law of sines. The law of sines states that in a triangle, the ratio of the length of a side to the sine of its opposite angle is constant.

In this case, the sides are the magnitudes of the forces, and the opposite angles are the angles between the forces and the positive x-axis.

Applying the law of sines, we have: (sin θ) / 500 = (sin 60°) / Resultant force

Solving for θ, we find that sin θ is equal to (sin 60°) / (Resultant force / 500). Calculating this equation, we get sin θ is approximately 0.250.

Taking the inverse sine of this value, we find that θ is approximately 14.6°. Therefore, the direction of the resultant force is approximately 14.6° with respect to the positive x-axis.

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115 degrees are both measure of variable r and t respectively

The given diagram is a triangle and we need to determine the measure of the unknown angles.

Since the angle is isosceles, the base angles will be equal, hence;

t = 180 - 65

t = 115 degrees

In the same way,

r = 180 - 65

r = 115 degrees

Hence the measure of r and t are 115 degrees.

In conclusion,  an isosceles triangle is described as  a triangle that has two sides of equal length.

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The equation which gives solutions, x=0 and y=-6.

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides. LHS = RHS is a common mathematical formula.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given:

y=-x-6............................(1)

x+3y=-18.........................(2)

now, solving both equation we get

x+3y=-18

x + 3(-x-6) = -18

x - 3x - 18 = -18

-2x = 0

x= 0

and, y= -0 -6 = -6

Hence, the solution is x=0 and y=-6.

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Check the picture below.

\((8+16)(8)=(12+x)(12)\implies 192=144+12x \\\\\\ 48=12x\implies \cfrac{48}{12}=x\implies 4=x\)

Answer:

x ≥ 3

Step-by-step explanation:

Answer:

10

Step-by-step explanation:

Rationalize the denominator by multiplying both the top and bottom by sqrt(5). you have sqrt(500) * sqrt(5) / sqrt(5)^2

simplifying we have 50/5 = 10