An 8-foot-long board is leaning against a building. The angle formed by the ground and the board is 70 ∘ . What is the distance from the building to the base of the board? Round to the nearest tenth.

There are several strategies for resolving systems of linear equations, including:

(1) Graphing Techniques

(2) The Substitution Method

(3) Using a linear combination

(4) The Matrix Approach

(1) Graphing Techniques: This is helpful if you only require a general answer or are confident that the intersection occurs at integer locations. Simply plot the two lines to determine their intersection.

(2) The Substitution Method: One linear equation for y must first be solved in terms of x. Then, in the other linear equation, replace y with that expression. You'll receive a formula in x. If you can solve this, you'll know the intersection's x-coordinate. Then, to determine the equivalent y-coordinate, insert x into either equation.

(3) Using a linear combination: In order for the x-terms or the y-terms to cancel out, add (or subtract) a multiple of one equation to (or from) the other equation. To obtain the other coordinate, first solve for x (or y, depending on which is remaining) and then substitute.

(4) The Matrix Approach: Actually, this is only a simplified notation of the Linear Combination method.

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The slope of a line perpendicular to 3x + 2y = 6 and passes through the (2,3) is 2/3.

The slope-intercept version of the line's equation is what we need. In other words, we want y=mx+b in the form of ax+by=c. M is the slope and b is the y-intercept in the slope-intercept form.

The slope of the line parallel to the provided line is equal to the reciprocal of the supplied slope in the negative direction. Therefore, the slope is m=-1/m. The original equation is solved to produce the following result in general form:

Equation of line intercept form is

y=mx+c

3x+2y=6

2y=6-3x

y = [tex]\frac{6-3x}{2}[/tex]

y = [tex]\frac{3-3x}{2}[/tex]

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

the slope of the line is m = -3/2

Slope of line which is perpendicular to the given line = 2/3

The new line passes by (2,3)

Equation of new line is y = 2/3x + 3.

So, the slope of a line perpendicular to 3x + 2y = 6 and passes through the (2,3) is 2/3.

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The complete question is:

What is the slope of a Line perpendicular to 3x + 2y = 6 and passes through the (2,3)?

The value of constant of proportionality is, 9,400

Any relationship that is always in the same ratio and quantity which vary directly with each other is called the proportional.

Given that;

The local dam lets 75,200 gallons pass through it over the course of 8 hours on Wednesday.

And, On Thursday, the dam lets 141,000 gallons pass through it over the course of 15 hours.

Now, We know that;

By the definition of proportionality,

⇒ y = kx

Where, k is constant of proportionality.

Let y represent the number of gallons and x represent the number of hours.

So, We get;

For The local dam lets 75,200 gallons pass through it over the course of 8 hours on Wednesday.

⇒ y = kx

⇒ 75,200 = k × 8

⇒ k = 9,400

And, On Thursday, the dam lets 141,000 gallons pass through it over the course of 15 hours.

⇒ y = kx

⇒ 141,000= k × 15

⇒ k = 9,400

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Answer:

x - 75 ≥ 420

Step-by-step explanation:

The domain of the function f(x) = (3 - 2x)/(9 - 4x) is equal to (-∞, 9/4) U (9/4, ∞) or (-∞, 2.25) U (2.25, ∞).

In Mathematics, a domain simply refers to the set of all real numbers for which a particular function is defined.

In order to determine the domain of this rational function, we would use an online graphing calculator to plot it and then write its domain in interval notation based on the values on the x-coordinate of the graph as follows:

Domain = (-∞, 9/4) U (9/4, ∞) or (-∞, 2.25) U (2.25, ∞)

By using set builder notation, the domain of this rational function f(x) = (3 - 2x)/(9 - 4x) can be re-written as follows;

Domain = {x|x ≠ 9/4} or {x|x ≠ 2.25}.

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The dimensions that will minimize the cost of the metal to manufacture the can is r = 5.42 dm and h = 10.84 dm.

To minimize the cost of the metal to manufacture the can, minimize the surface area of the can. The surface area of a cylinder can be found using the formula 2πr(r + h) where r is the radius and h is the height.

The volume of the cylinder is given by the formula πr²h, and it is known that it needs to be equal to 1000 l or 1000 cubic dm.

Set the volume equation equal to 1000 and rearrange it in terms of r.

πr²h = 1000 cubic dm

h = 1000/(πr²)

Substitute this value of h back into the surface area equation and minimize it with respect to r.

2πr(r + h) = 2πr(r + 1000/(πr²))

To minimize the surface area, take the derivative of this expression with respect to r and set it equal to zero, then solve for r.

d(2πr(r + 1000/(πr²)))/dr = 0

d(2πr² + 2000πr/(πr²))/dr = 0

d(2πr² + 2000/r)/dr = 0

4πr - 2000/r² = 0

4πr = 2000/r²

r³ = 2000/4π

r = ∛(500/π)

r = 5.42 dm

Substitute it back into the equation for h and find the corresponding value of h.

h = 1000/(πr²)

h = 1000/(π(∛(500/π))²)

h = 10.84 dm

So, the dimensions that will minimize the cost of the metal to manufacture the can are r = 5.42 dm and h = 10.84 dm.

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hi there! the slope for those two points is 10/5

The diagonal length of the fence is 89.44 units.

Square is a polygon which has 4 sides. All four sides of the square are equal length and perpendicular to each other, that means the angle between two adjacent side is 90°.

Given:

The area of the square field is 40,000 squared unit.

We want to but a fence diagonally through the field.

To find the diagonal:

The diagonal of square,

d=a√2, where d is the diagonal and a is the side of the square.

d = √(40000)√2

d = √(80000)

d = 89.44.

Therefore, the length of the fence is 89.44 units.

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The required  temperature of the oven 17 and t minutes after being turned on are 245º and 75º + 17t.

The formula R = D/T, or rate of change equals the distance travelled divided by the time required to do so, can be used to approach rate of change problems in general.

The initial temperature of the oven is 75º and temperature will increase at a rate of 10° per minute after being turned on.

Total temperature after 17 minutes,

75º  + 17(10º )

75º + 170º

245º

And

Temperature after oven t minutes after being turned on

75º + 17t

Thus, required temperature of oven is 245º and 75º + 17t.

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The wingspan of the airplane in the scale drawing  is equal to 6.67 inches.

The wingspan of an airplane in a scale drawing can be calculated using the scale factor. If the wingspan of the real airplane is 60 feet and 1 inch represents 9 feet, then the wingspan of the airplane in the scale drawing is 60/9 = 6.67 inches. Knowing the dimensions of the real airplane and the scale factor of the drawing, it is possible to accurately calculate the wingspan of the airplane in the scale drawing. This calculation is important for accurately measuring and drawing scale models of aircraft.

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The total interest that Gina will Earn on her deposit through the end of the fifth year is, $32.4.

The simple interest formula is I = Prt, where P is the principal (original amount of deposit), r is the interest rate, and t is the time in years.

In Gina's case, the principal is $150, the interest rate is 4%, and the time is 5 years.

To find the interest earned in the first year, we can plug the values into the formula:

     I = 150(0.04)(1) = $6

To find the interest earned in the second year, we calculate the interest on the original deposit plus the interest earned in the first year:

     I = (150 + 6)(0.04)(1) = $6.24

To find the interest earned in the third year, we calculate the interest on the original deposit plus the interest earned in the first and second years:

     I = (150 + 6 + 6.24)(0.04)(1) = $6.48

To find the interest earned in the fourth year, we calculate the interest on the original deposit plus the interest earned in the first, second and third years:

    I = (150 + 6 + 6.24 + 6.48)(0.04)(1) = $6.72

To find the interest earned in the fifth year, we calculate the interest on the original deposit plus the interest earned in the first, second, third and fourth years:

    I = (150 + 6 + 6.24 + 6.48 + 6.72)(0.04)(1) = $6.96

To find the total interest earned through the end of the fifth year, we add up the interest earned in each year:

   6 + 6.24 + 6.48 + 6.72 + 6.96 = $32.4

Therefore, Gina will earn a total of $32.4 in interest on her deposit through the end of the fifth year.

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The given line segment which is divided into the ratio 3 : 4  shows line is divided into 7  parts.

As given in the question,

Let us consider the given line segment be PQ .

And take point R between the given points AB such that it get divided in the ratio 3 : 4.

From the given ratio PR is divided into 3 equal parts.

And RQ  is divided into 4 equal parts.

Total number of parts PQ  is divided is given by

= PR + RQ

= 3 + 4

= 7  parts  which are equal to each other.

Therefore,  the given ratio 3 : 4 represents that the line segment is divided into 7 equal parts.

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Between these two integers the square roots lie

a)√ 56=7 < √56 < 8

b)-√78= 8 < -√78 < 9

c) -√43=-7 < -√43 < -6

d)√99= 9 < √99 < 10

Find the square of the two integers that are immediately adjacent to the square root. For instance, we would discover the square of 7 and 8, which is 49 and 64, respectively, for 56.

The surd should be compared to the square of the adjacent numbers. The surd is said to be between two integers if it is bigger than the square of the lower integer and less than the square of the higher integer.

This approach allows us to see that √56 is between 7 and 8 as 49 < √56 < 64.

Similarly,

b) -√78 lies between 8 and 9 because 64 < -√78 < 81

c) -√43 lies between -7 and -6 because 49 > -√43 > 36

d) √99 lies between 9 and 10 because 81 < √99 < 100

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The system if there are fewer equations than variables has infinite solutions.

There are two types of system of equations - consistent and inconsistent.

Inconsistent means that it has no solution , i.e. the solution does not exist.

Consistent system means a solution of the equation exists.

Now, a consistent system may have a unique solution or an infinite number of solutions where we need to assign an arbitrary value to a free variable to find its solutions.

For e.g. let us consider the system -

x + y+ z = 0

2x + 3y + 4z = 1

Since , (0,0,0) is an obvious solution to this system , its a consistent system .

Also, for a system to be consistent , either a unique solution exists or an infinite number of solutions exist. There is no particular number of solutions.

Also, we see that (-1,1,0) is also a solution other than the zero solution.

We can clearly see that the number of unknown variables , x,y,z is 3 and the number of equations is 2.

Thus, The system if there are fewer equations than variables has infinite solutions.

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The 7 rules of logarithms are as: 

1. Rule of product.

2. Rule of division.

3. The Power/Exponential Rule.

4. A new fundamental rule.

5. Rule of the base swap.

6. Log derivative.

7. Log integral.

The alternate format for writing exponents is with logarithms. A number appears base-based logarithm is approximately equivalent to another number. A logarithm performs exponentiation's exact opposite function.

1. Rule of product: According to this rule, adding the individual logarithms of two logarithmic values results in the multiplication of those values.

[tex]Log_b (mn)= log_b m + log_b n[/tex]

2. Rule of division: The difference between each logarithm is equivalent to the division of two logarithmic values.

[tex]Log_b \left(\frac{m}{n}\right)= log_b m - log_b n[/tex]

3. The Power/Exponential Rule: According to the exponential rule, the exponent times the logarithm of m's logarithm is equivalent to m's logarithm with a reasonable exponent.

[tex]Log_b (m^n) = n log_b m[/tex]

4. A new fundamental rule:

[tex]Log_b m =\frac{log_a m}{log_a b}[/tex]

5. Rule of the base swap:

[tex]log_b (a) = \frac{1}{log_a (b)}[/tex]

6. Log derivative: If f (x) = [tex]log_b (x)[/tex], then the derivative of f(x) is given by;

[tex]f'(x) = \frac{1}{(x ln(b))}[/tex]

7. Log integral:

[tex]\int{log_b(x)dx} = x\left(log_b(x) - \frac{1}{ln(b)}\right) + C[/tex]

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It will take 3750 minutes (62.5 hours) time for all the water to flow out of the tank.

1. Calculate the total volume of the tank: 4500 litres

2. Calculate the rate of flow: 1.2 litres per second

3. Calculate the total time it will take for all the water to flow out: 4500 / 1.2 = 3750 minutes (62.5 hours)

The tank contains 4500 litres of water and is flowing out of the tank at a rate of 1.2 litres per second. This means it will take 3750 minutes (62.5 hours) for all the water to flow out of the tank. To calculate this, we first had to determine the total volume of the tank which was 4500 litres. Then, we had to calculate the rate of flow which was 1.2 litres per second. Finally, using the total volume of water and the rate of flow, we were able to calculate that it would take 3750 minutes (62.5 hours) time for all the water to flow out of the tank.

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The statement angle 8=85, because 180-115=65 , 180-(65+30)=85 is true.

A triangle is a three-sided polygon that consists of three edges and three vertices.

In the given triangle one measure is given which is 30 degrees.

The angle 7 we calculate because the other side angle is 115 degrees.

The sum should be 180 degrees

115+angle 7=180

angle 7=180-115

angle 7=65 degrees

Now we know that the sum of interior angles of a triangle are 180 degrees.

65+30+angle 8=180

95+angle 8=180

Angle 8=85

Hence, the statement angle 8=85, because 180-115=65 , 180-(65+30)=85 is true.

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Answer:

16/52

8/26

4/13

As 13 cards are one suit or are hearts, and in that suit, there is one king, I just add 3 more cards for the other king cards as well and you get 16/52

Step-by-step explanation:

The relationship between two equal amounts is described by the characteristics of equality

The relationship between two equal amounts is described by the characteristics of equality. To maintain balance, each mathematical operation that is done to one side of an equation must also be applied to the opposing side.

a = 2b, then a-c = 2b - c

This is Subtraction Property of Equality

According to the Subtraction Property of Equality, if two equal things have an equal value subtracted from them or deleted, the outcome is a new equal quantity. When two equivalent sides of a mathematical statement are separated by an equal sign, an equation is being expressed. The subtraction property has the expression if a = b, then a - c = b - c.

X = X

This represents the Reflexive Property

In algebra, a number is always equal to itself according to the reflexive property of equality.

3(p-7) = 3p - 21

This represent distributive property.

This property states that multiplying the total of two or more addends by a number will provide the same outcome as multiplying each addend by the number separately and then adding the results together.

-7k = -42 then k= 6

This is Division Property of Equality

According to the division property of equality, the quotients of an equation remain equal when both sides are divided by a common real integer that is not equal to 0. The same formula may be expressed in writing as follows: If real numbers a, b, and c are present, with a = b and c 0, then a c = b c.

m + n = 15 and n = 2, then m + 2 = 15

This represent Substitution Property

x/4 = -5 then x = -20

This is Multiplication Property of Equality

w² = 2x and 2x = y, then w² = y

This is Transitive Property

c-9 = -1, then c = 8

This is Addition Property of Equality

n = -3, then -3 = n

This is Symmetric Property

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Therefore , the solution of the given problem of inequality comes out to be x<6 and x>9.

A connection of linear solutions or integers in mathematics that lacks an equal sign is known as an inequality. Inequity so follows an imbalance. There exists an inequality between two numerical values when they are not equal. Differences exist between equality and inequality. I've used the most typical symbol in cases where the variables aren't comparable or equal (). Regardless of how little or large the values are, there are numerous inequities that can be used to compare them. A variety of fundamental inequality issues can be solved by switching the two sides until variables represent everything that is left. There are numerous variables that could lead to inequality, including the conjunction or division of low values on both sides. Alternating sides

Here,

Given : 5x + 2 < 32

=> 5x < 32-2

=>5x < 30

=>x < 6

thus,

=> x-7>2

=> x>9

Therefore , the solution of the given problem of inequality comes out to be x<6 and x>9.

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The solution to the system of linear equation x, y are -6, 6

A system of linear equations is a collection of two or more linear equations that contain the same variables. Linear equations are equations of the form ax + by = c, where a, b, and c are constants and x and y are variables.

There are different methods to solve a system of linear equations, such as substitution, elimination, and graphing. Each method has its own advantages and limitations, and the choice of method depends on the form of the equations and the type of solution.

In this problem, we have two system of equations which are given as;

Combining both equations and solving via substitution method, the values of x and y are -6 and -6

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Answer:

B) m × 3 = 27

Step-by-step explanation:

Explanation:

m = 27 ÷ 3 = 9

E) q ÷ 3 = 27

Explanation:

q = 27 × 3 = 81

So, the equations with 9 as a solution are B and E.

On solving the provided question we can say that equation will be as follows  6x + 55y = 13.15 => x = coins = 2 and y = nickels = 12

An equation is a mathematical formula that connects two assertions using the equal sign (=) to denote equivalence. In algebra, an equation is a mathematical statement that establishes the equivalence of two mathematical expressions. For instance, an equal sign separates the components 3x + 5 and 14 in the equation 3x + 5 = 14. A mathematical formula is used to explain the connection between two sentences on either side of a letter. Frequently, there is just one variable, which is also the symbol. for example, 2x - 4 = 2.

jesse has  = $6.55 worth of quarters and nickels

the number of nickels is five more than two times the number of quarters

so

let coins be x

and nickels be y

now equation will be as follows

6x + 55y = 13.15

x = coins = 2

y = nickels = 12

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There are 126 different selections possible if at least 3 balls must be blue.

Total number of red balls: 8

Total number of blue balls: 8

Also, a woman selects 4 balls

And also at least 3 balls off the 4 must be blue

So the first combination is 3 blue and 1 red remember the woman picks up 4 balls.

And the second combination is 4 blue and 0 Red remember at least 3 blue balls

So the equation is as given below:

[tex]^nC_{r} =\frac{n(n-1)(n-2)(n-3)..... r terms.}{1*2*3..... r terms.}[/tex]

Now substitute values in the formula:

(8C3*8C1 + 8C4*8C0)=(56+70)=126 selections.

Hence, There are 126 different selections are possible if at least 3 balls must be blue.

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Mountain View received at least 4000 visits 66.67% of the time during their busy season.

Calculating the percentage of times Mountain View received at least 4000 visitors is the task at hand in this exercise. According to the data plot, we can see that they had no visitors the first weekend, 2000 the following weekend, 4,000 the third week, 6,000 the fourth, 8,000 the following week, and 10,000 the last week of their busy season. We can see that starting in week 3, they had more than 4,000 visits, making the total number of weeks within our 6-week study period that had more than 4,000 views equal to 4 weeks. Therefore, (4/6)*100, or about, 66.67% represents the percentage of visitors of at least 4,000 at Mountain View.

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His total profit is  13130. Your remaining cash balance (also known as your net profit) is the sum of all your business expenses.

The answer is given below :

Sum = Rs 40000

We know that,

Interest = (Sum × Time × Rate)/100

So, for the 1st year when profit is 5%,

Interest

= Rs (40000 × 1 × 5)100

= Rs 2000

So, according to the question,

Rs (40000 + 2000) = Rs 42000 be the Sum for the 2nd year.

So, for the 2nd year, when profit is 10%,

Interest

= Rs (42000 × 1 × 10)/100

= Rs 4200

Again, for the 3rd year, Sum

= Rs (42000 + 4200)

= Rs 46200

Thus, for the third year, when profit is 15%,

Interest

= Rs (46200 × 1 × 15)/100

= Rs 6930

So, now amount is total

= Rs (46200 + 6930)

= Rs 53130

Hence, total profit

= Rs (53130 - 40000)

= Rs 13130

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Answer:

B

Step-by-step explanation:

B.B.B.B.B.B.BB.B.B.B.B.B.B.B.B

Answer:y=2x+5

Step-by-step explanation:

y intercept of 5

the distance between the points on x axis is 4

the distance between the points on y axis is 8

8/4=2

gradient of 2

Answer:

2x+5 = two will divide five which is 2.5 so y=1/2 +2.5x makes 5

Step-by-step explanation:

Answer =2x+5

The inverse of the logarithmic function 3 is 1000 for base 10.

The index or power to which the base must be raised in order to reach the given number is represented by the logarithm of any positive number, whose base is a number that is larger than zero but not equal to one.

mathematical form: [tex]log_ab=x[/tex].

Here for inverse of given function is derived by given formula.

[tex]a^x=b[/tex]

There are two types of logarithm function:

Natural Logarithm and common logarithm.

[tex]log_{a}b=x\\log_{10}b=3\\a^x=b\\10^3=b\\b=1000[/tex]

So inverse of logarithmic function 3 is 1000.

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