Similar triangles: To indirectly measure the distance across a river, Madeline stands on one side of the river and uses sight-lines to a landmark on the opposite bank. Madeline draws the diagram below to show the lengths and angles that she measured. Find PR, the distance across the river. Round your answer to the nearest foot.

Check the picture below.

[tex]\textit{perimeter of a rectangle}\\\\ P=2(L+w) ~~ \begin{cases} L=length\\ w=width\\[-0.5em] \hrulefill\\ P=88\\ L=w+8 \end{cases}\implies 88=2( ~~(w+8)+w ~~ ) \\\\\\ 88=2(2w+8)\implies 88=4(w+4)\implies \cfrac{88}{4}=w+4 \\\\\\ 44=w+4\implies \boxed{40=w}\hspace{5em}\stackrel{w+8}{\boxed{L=48}}[/tex]

The side length of a square with area of 7 units squared is given as follows:

2.65 units.

The area of a square of side length s is given by the side length squared, according to the equation given as follows:

A = s².

In this problem, we have that the output is of 7, that is, the area of the square is of seven, hence the side length of the square is given as follows:

s² = 7

s = sqrt(7)

s = 2.65 units.

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The coordinates of the position at which Pablo should stand are given as follows:

(6, 7.5).

The midpoint between two points is the halfway point between them, and is found using the mean of the coordinates.

Hence the x-coordinate of the position at which Pablo should stand is given as follows:

x = (2 + 10)/2

x = 6.

The y-coordinate of the position at which Pablo should stand is given as follows:

y = (3 + 12)/2

y = 7.5.

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Answer: (21, 3)

Step-by-step explanation:

x is a component of the real number system 11 < 3x-2 ≤ 19, although as a compound inequality, it also belongs to 13/3 and 7.

A interaction between two terms or concepts that is neither equal in mathematics is referred to as an inequality. Consequently, inequity results from imbalance. In algebra, an inequality establishes the connection between two non-equal numbers. Egality and inequalities are not the same. Use the not comparable symbol most usually when two components are really not equal (). Values of any size can be contrasted using a variety of inequalities. By changing the two aspects until only the variables are left, many straightforward inequalities can be solved. But, a variety of factors support inequality: Both sides' 0 are split or added. Exchange the left and the right.

given

11 < 3x-2 ≤ 19

11< 3x-2

3x-2 ≤ 19

11+ 2 < (3x-2)+ 2

(3x-2)+ 2 ≤ 19 + 2

x belongs to 13/3 and 7

x is a component of the real number system 11 < 3x-2 ≤ 19, although as a compound inequality, it also belongs to 13/3 and 7.

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The logical equivalence for the given statement is (p → r) ↔ (q → r)≡ (~p∨r)∧(~q∨r).

Logical equivalence is the condition of equality that exists between two statements or sentences in propositional logic or Boolean algebra.

Given that, p → q ≡ ~p∨q and p ↔ q ≡ (~p∨q)∧(~q∨ p)

a) (p → r) ↔ (q → r)

Now, (p → r)≡ ~p∨r

(q → r)= ~q∨r

So, (p → r) ↔ (q → r)≡ (~p∨r)∧(~q∨r)

Therefore, the logical equivalence for the given statement is (p → r) ↔ (q → r)≡ (~p∨r)∧(~q∨r).

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Answer: 485 is the answer.

Step-by-step explanation:

The given triangle CDE is not Rectangular.

What are types of triangles ?

Acute triangle . . . all three angles are less than 90°

Right triangle . . . one angle is 90°

Obtuse triangle . . . one angle is greater than 90°

Equilateral triangle . . . all three sides are the same length

Isosceles triangle . . . two sides are the same length

Scalene triangle . . . all three sides are different lengths

Given:

CD = 36cm ; CE = 42cm and DE = 55

Hence , the given triangle CDE is not Rectangular.

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The values of the composite functions are g(f(0))= -3 and f(f(2)) = 5

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

f(x)= 2x-1 g(x)=3x and h(x)=x^2+1

To calculate g(f(0)), we have

f(0)= 2(0)-1 = -1

This means that

g(f(0)) = g(-1)

So, we have

g(f(0)) = 3(-1) = 3

For f(f(2)), we have

f(2)= 2(2)-1 = 3

So, we have

f(f(2)) = 2(3) - 1 = 5

Hence, the composite functions are g(f(0))= -3 and f(f(2)) = 5

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Answer:

(4.44 + 0.55n) / n = p

(4.44 + 0.55 * 6) / 6 = p

(4.44 + 3.3) / 6 = p

(7.74) / 6 = p

1.29 = p

or

4.44 / 6 + 0.55 = p

Step-by-step explanation:

n = number of bottles

p = 1.29

-22 < 7x is the inequality that is equivalent to the given inequality.

What is inequality?

An inequality is a relation that compares two numbers or other mathematical expressions in an unequal way. The majority of the time, size comparisons between two numbers on the number line are made. Different types of inequalities are represented by a variety of notations, including:

The symbol a < b indicates that an is smaller than b.

When a > b is used, it indicates that an is bigger than b.

Here, we have

Given: -4(x + 7) < 3(x - 2)

To find out which inequality is equivalent, you solve for x.

-4(x+7) < 3(x-2): distribute variables

-4x-28 < 3x-6: Add 6 to both sided

-4x-22 < 3x: add 4x to both sides

-22 < 7x

Hence,  -22 < 7x is the inequality that is equivalent to the given inequality.

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Answer: Yes.

Step-by-step explanation: Follow PEMDAS.

P: PARENTHESIS

E: EXPONENTS

M: MULTIPLY

D: DIVIDE

A: ADDITION

S: SUBTRACT

It cannot be proven that the front faces are congruent triangles.

No, I cannot prove that the front faces of the two pyramids of Giza are congruent triangles. The exact dimensions and shapes of the pyramids have been lost to history, and the information we have today is based on estimates and interpretations of archaeological evidence.

The exact shape of the pyramids is still the subject of some debate among experts, and there is no consensus on whether the front faces are congruent triangles.

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The fraction pairs that are equivalent to 4 and 20 is given as follows:

Equivalent fractions are fractions that have the same denominator.

The fraction for this problem is given as follows:

4/20.

Dividing both the numerator and the denominator by 2, an equivalent fraction is given as follows:

2/10.

Dividing again by two, we have that:

1/5.

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The sample size based on the information is 30.

The probability is 0.1667.

From the information, Mr. Jackson could think about his experiment a little differently. How many times should he expect to roll a 4 if he rolls his number cube 30 times? This is called the mean, or expected value.

The Sample size , n = 30. The Probability of an event of interest, p = 0.1667

The Mean will be:

= np = 30 * 0.167

= 5.000

Standard deviation will be:

= √(np(1-p)) = √ 4.1667

= 2.04

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The value of the equations are

A =  0.93 x 2.6 = 2.418

B = 93 x 0.26 = 24.18

C = 9.3 x 26 = 241.8

D = 0.93 x 0.26 = 0.2418

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.

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 data ,

Given data ,

Let the equation be represented as A

Now , the value of A is

Substituting the values in the equation , we get

a)

A = 0.93 x 2.6   be equation (1)

On simplifying the equation , we get

A = 2.418

b)

B = 93 x 0.26   be equation (2)

On simplifying the equation , we get

B = 24.18

c)

C = 9.3 x 26   be equation (3)

On simplifying the equation , we get

C = 241.8

d)

D = 0.93 x 0.26   be equation (4)

On simplifying the equation , we get

D = 0.2418

Hence , the equations are solved

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The fifth term of the geometric progression is given as 1280

The formula for calculating the nth term of a geometric progression is expressed as:

Tn = ar^n-1

Given the following

First term = a = 5

Third term = ar^2 = 80

Determine the common ratio

5(r^2) = 80
r^2 = 16

r = 4

For the 5th term

T5 = ar^4

T5 = 5(4)^4
T5 = 1280

Hence the fifth term of the expression is 1280

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Complete question

The first and third term of a Gip are 5 and 80 respectively. What is the fifth term​

Problem 1

L = length

W = width

P = perimeter of the rectangle

P = 2L+2W

P = 20

2L+2W = 20

L+W = 10 after dividing everything by 2.

L = 10-W

area = length*width

A = L*W

A = (10-W)*W

A = -W^2 + 10W

Let x replace W for a temporary basis.

If we graphed y = -x^2+10x, then we'd see the highest point occurs at (5, 25). This means a width of 5 meters leads to the largest area of 25 square meters.

The length would be L = 10-W = 10-5 = 5 meters as well.

It turns out that a 5 meter by 5 meter square is the best choice when trying to max out this rectangle's area.

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

Problem 2

The results we got from the previous section was that a 5 by 5 square will max out the area.

If we allowed non-whole numbers into the mix, then it wouldn't change the previous answer. The best choice is still that 5 by 5 square.

The answer only would change if the result of the previous section was some non-integer value.

Answer:

  ∆ABC is an isosceles triangle

Step-by-step explanation:

You have a triangle ABC such that the angle bisector of angle B also bisects side AC.

An angle bisector divides the sides of the triangle proportionally. This means the sides BA and BC will have the same ratio as the segments of AC on either side of the point of intersection with the angle bisector.

Since AC is bisected, that means those segments are congruent, hence sides BA and BC are congruent. The triangle is isosceles.

The required final balance on interest rate of 6%per annul is  $354.

The fee you pay to borrow money or the fee you charge to lend money is called interest. The most common way to represent interest is as a yearly percentage of the loan amount. The interest rate on the loan is known as this percentage.

Simple interest is calculated as:

I = P * r * t

where I is the interest, P is the principal (initial investment), r is the annual interest rate as a decimal, and t is the number of years.

So, in this case:

I = 300 * 0.06 * 3

I = 300 * 0.06 * 3 = 54

The final balance would be the sum of the initial investment and the interest earned:

Final balance = P + I

Final balance = 300 + 54

Final balance = 300 + 54 = 354

Rounded to the nearest cent, the final balance is $354.

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Answer:

4 feet and 4 inches of fringe

Step-by-step explanation:

The length and width of a square are equal to each other

Let x = length of a side of a square

Area of a square = Length × Width

                            = (x inches) × (x inches)

                            = [tex]x^{2}[/tex] [tex]in^{2}[/tex]

The area of the square placemat = 169 [tex]in^{2}[/tex]

∴                                            [tex]x^{2} in^{2}[/tex]  = 169 [tex]in^{2}[/tex]

Taking the square root on both sides of the equation to get rid of the square:

                                             x inches = [tex]\sqrt{169}[/tex] inches

                                            x inches = [tex]\sqrt{169}[/tex] inches

   ([tex]\sqrt{169} =[/tex] ± 13. However, the negative value will be rejected since the length of a side CANNOT be negative)    

        So, x = 13 inches

This means the perimeter of the square = (13 + 13 + 13 + 13) inches

                                                                   = 52 inches

∴ 52 inches of fringe will be needed by Sally

Unit conversion:

                                                   I foot = 12 inches

                                      ∴          y feet = 52 inches

     Cross-multiplication is applied:

                       (y feet)(12 inches) = (52 inches)(1 foot)

Isolating y and making it the subject of the formula:

                                   y feet = [tex]\frac{(52 inches)(1 foot)}{12 inches}[/tex]

Inches in the numerator and denominator will cancel each other out completely:

                                  y feet = [tex]\frac{52}{12}[/tex] feet

Applying long division:

The highest quotient to be multiplied by 12 to give a number, which is closest to 52 but still less than 52. 12×4 = 48. Therefore, the quotient is 4 and the remainder is 4.

∴ 52 inches = 4 feet and 4 inches

Elisa was charged a basic interest rate of 120% per year by the pawnbroker since she needed the money to fix her house air conditioner.

By dividing the initial by the interest income, the aging process, and other factors, simple interest is determined. Simple yield equals principal plus interest full days is the concept used in marketing. The easiest thing to compute interest is to employ this method. The most frequent method for calculating it is as a portion of the outstanding sum. For example, if he takes $100 from a buddy and undertakes to return the loan at 5% interest, he will only spend his share of the 100% interest. $100 (0.05) = $5. When you borrow money, you must make payments, and you must pay the interest whenever you lend it. Typically, interest is computed as an indicator of the overall of the loan total. The loan's interest is the name given to this percentage.

given

I = PRT/100

Simple interest (I) is calculated as follows: amount - principal ($279.90 - $270) = $9.90

P (principal) is the $270 loan made to Elisa.

R is the interest rate on a yearly basis.

The loan's amortisation period, or T (time), is 11 days, or 11/360 years.

9.9 = 270 × R × 11/360 × 1/100

R = (9.9×360×100)/(270×11) = 120%

120% is the annual simple interest rate.

Elisa was charged a basic interest rate of 120% per year by the pawnbroker since she needed the money to fix her house air conditioner.

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The probability of F′∪A 0.9829

The probability can be defined as the likelihood that an event is going to happen.

We are to find the probability F′∪A.

What is F'

This is gotten as 1 - probability of F

F' = 1 - 0.2058

F' = 0.7942

Then we are to get the probability of event A from the question

P(A) = 0.1887

Hence the probability of F′∪A.

= P(A) + P(F`)

= 0.1887 + 0.7942

= 0.9829

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The probability that the test is negative for all people is: P = 0.9655 and probability that the test comes back positive for at least one of the five people is P = 0.0344.

Probability refers to potential. A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has incorporated probability to forecast the likelihood of various events. The degree to which something is likely to happen is basically what probability means. You will understand the potential outcomes for a random experiment using this fundamental theory of probability, which is also applied to the probability distribution. Knowing the total number of outcomes is necessary before we can calculate the likelihood that a specific event will occur.

The probability that the test is negative for all people is:

[tex]P = (0.995)^7[/tex]

P = 0.9655

b. The probability that the test comes back positive for at least one of the five people:

P = 1 - P (Negative for all 5)

P = 1 - 0.9655

P = 0.0344

Hence, the probability that the test is negative for all people is: P = 0.9655 and probability that the test comes back positive for at least one of the five people is P = 0.0344.

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According to the question we have ,The area covered by the spotlight is 153.86 square feet. The correct option is c.

The diameter of a circle cuts through center whereas the radius extends from the center to the edges of the circle. A circle's diameter effectively splits the shape in half. The radius of a circle is the distance from the middle to any point on the edge. The diameter the radius are inversely proportional, or 2r=d2 r=d. A line connecting two points on either a circle is referred to as a chord.

We have the formula of Area of a circle: A = π[tex]r^2.[/tex]

According to the question we have ,there is a spotlight above the center of the ground that surrounds a 7-foot-radius lit area.

A = π[tex]r^2[/tex]. = π * [tex]7^2[/tex]

= 3.14 * 49 = 153.86 square feet.

So, the area covered by the spotlight is 153.86 square feet.

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Answer: To find the value of w, we need to set the expressions equal to each other:

(4w-8) = (3w+6)

Subtracting 3w from both sides:

w - 8 = 6

Adding 8 to both sides:

w = 14

So the value of w is 14.

Step-by-step explanation:

Answer:

Step-by-step explanation:The equation for the nth term of an arithmetic sequence is given by:

an = a1 + (n-1)d

Where:

a1 is the first term of the sequence

d is the common difference

n is the number of the term we want to find

Using the given information, we have:

a1 = -9 and d = 2

Plugging these values into the formula, we get:

an = -9 + (n-1)2

To find the 12th term of the sequence:

n = 12

an = -9 + (12-1)2

an = -9 + 11 * 2

an = -9 + 22

an = 13

So the 12th term of the sequence is 13.