using the digits $1$, $2$, $3$, $4$, $5$, how many even three-digit numbers less than $500$ can be formed if each digit can be used more than once?

Step-by-step explanation:

Sin, Cos, and Tangent are the three common trig functions.

Sine, when given an angle, is a right triangle ratio between the opp side of the angle /hypotenuse(the side opposite of the right triangle),

SOH

Cosine, when given an angle, is a right triangle ratio between the adj side of the angle/hypotenus(the side opposite of the right triangle)

CAH

Tangent, when given an angle, is a right triangle ratio between the opp side of the angle/ adj side of the angle

TOA.

So

given an angle, alpha,

[tex] \sin( \alpha ) = \frac{o}{h} [/tex]

[tex] \cos( \alpha ) = \frac{a}{h} [/tex]

[tex] \tan( \alpha ) = \frac{o}{a} [/tex]

Problem 1, we know that the angle of reference is 53.1, the side adjacent to that angle is 30 cm, and we are trying to find x. and y.

Solving for x, the side length x is opposite of the angle 53.1

We have an angle, the adjacent side of the angle and the opposite side is what we are trying to find.

We use TAN

[tex] \tan(53.1) = \frac{x}{30} [/tex]

We know

[tex] \tan(53.1) = 1.33[/tex]

so

[tex]1.33 = \frac{x}{30} [/tex]

[tex]40 = x[/tex]

Now, since we know the side length of x, we have two ways to solve for y, which is the hypotenuse.

Remember the hypotenuse is the angle opposite of the 90° angle(right angle)

We could use Sine or Cosine

[tex] \sin( 53.1) = \frac{40}{y} [/tex]

[tex]0.8 = \frac{40}{y} [/tex]

[tex]y = \frac{40}{0.8} [/tex]

[tex]y= 50[/tex]

or

[tex] \cos(53.1) = \frac{30}{y} [/tex]

[tex]0.6 = \frac{30}{y} [/tex]

[tex]y = 50[/tex]

So x=40, y=50

2. Do the same steps above

Solving for x,

[tex] \sin(30) = \frac{x}{90} [/tex]

[tex]45 = x[/tex]

Solving for y,

[tex] \cos(30) = \frac{y}{90} [/tex]

cos(30)

[tex] \cos(30) = \frac{ \sqrt{3} }{2} [/tex]

so

[tex] \frac{ \sqrt{3} }{2} = \frac{y}{90} [/tex]

[tex]y = 45 \sqrt{3} = 77.942[/tex]

The lengths of the three sides are 15 inches , 19 inches and 25 inches respectively.

Let a represent the length of the shortest side.

Let b represent the length of the mid-length side.

Let c represent the length of the longest side.

From the question, we have

The perimeter of a triangle is 59 in.

a + b + c = 59

The shortest side is 10 in. less than the longest side.

a = c-10

The middle side is 21 in. less than the combined lengths of the shortest and longest sides.

b=(a+c)-21

b=(59-b)-21

2b = 38

b = 19 inches

substituting the value  we get

a+c = 40................(i)

a-c = -10................(ii)

Solving, we get

2a = 30

a = 15 inches

c=25 inches

The lengths of the three sides are 15 inches , 19 inches and 25 inches respectively.

Addition:

The act of combining two or more items is known as addition. The process of computing the sum of two or more numbers in mathematics is called addition. It is a fundamental mathematical procedure that is frequently used in daily life. When we work with money, figure out our food bills, or figure out the time, we frequently employ addition. One-digit numbers can be added simply for solving addition problems, however for bigger numbers, we divide the numbers into columns using their corresponding place values, such as ones, tens, hundreds, thousands, and so on.

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The average velocity for the time period is -15 feet per second

The velocity of the ball = 31 feet / seconds

The function of the height is

y = 31t - 23t^2

Time period beginning when t = 1 and lasting .01 s, .005 s, .002 s, .001 s

The average velocity = f(1.01) - f(1) / 1.01 - 1

f(t) = 31t - 23t^2

f(1.01) = 31(1.01) - 23(1.01)^2

= 31.31 - 23.46

= 7.85 feet per second

Similarly

f(1) = 31(1) - 23(1)^2

= 31 - 23

= 8 feet per second

The average velocity =  7.85 - 8 / 1.01 - 1

= -0.15 / 0.01

= -15 feet per second

Hence, the average velocity for the time period is -15 feet per second

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The total amount the college student has in his saving account if 13000 dollars  was invested and earned 3% compounded quarterly for 5 years    is  $15,095.39

The total amount a college student has in a saving account if 13000 dollars was invested and earned 3% compounded quarterly for 5 years can be calculated as follows:

Therefore,

[tex]A = p(1 +\frac{r}{n} )^{nt}[/tex]

where

Therefore,

p = 13000 dollars

t = 5 years

n = 4

r = 3% = 0.03

Therefore,

[tex]A = p(1 +\frac{r}{n} )^{nt}[/tex]

[tex]A = 13000(1+ \frac{0.03}{4} )^{(4)(5)}[/tex]

A = 13,000.00(1 + 0.0075)²⁰

A = $15,095.39

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Answer:

Angle3 and Angle6

(also, Angle4 and Angle5)

Step-by-step explanation:

Alternate interior angles are between the parallel lines and also on opposite sides of the transversal.

The number of different combination may a new car would have is 10 ways.

What is a combination in mathematics?

The combination is defined as "an arrangement of objects in which the order in which the objects are chosen is unimportant." The combination means "selection of things," where the order is irrelevant.

Given that there are 10 different options.

We have to choose one car.

So total number of combinations = [tex]^{10}C_1[/tex]

                                                       = 10!/(9!)(1!)

                                                       = 10 ways

Hence, the number of different combination may a new car would have is 10 ways.

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In 10 ways the combination of a new car can be installed as followed by the rules of Permutation combination.

what is Permutation combination ?

Putting things or numbers in order is the definition of a permutation. Combinations are a method of choosing elements such as numbers or objects from a collection or set of elements without regard to the elements' order.

Solution:

Given that the car are 10 different options.

We have to choose one car.

So total number of combinations = 10C1

                                                      = 10!/(9!)

                                                      = 10 ways

Therefore, In 10 ways the combination of a new car can be installed.

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A 10% increase in the price of good Y will result in a 6% increase in the quality demanded of good X.

A number or variable is a quantity. Measuring the percentage change of the quantity demanded for a good to the percentage change in the price of another good is called cross-elasticity of demand.

As per the question,

As the cross-elasticity of the goods is 0.6 i.e. 1% will increase in the price of good Y, increasing the demand for good X by 0.6%. X and Y are the substitute goods because when the price of Y increases sales of X increase.

⇒ When the price of good Y increases by 10%, then that will result in

    increase in quantity demanded of good X   =   0.6 × 10

                                                                              =   6%

⇒ quantity demanded of good X is 10 units, 6% increase means

    quantity demanded of good X   =   ( 1.06 × 10 )

                                                          =   10.6 units

Therefore, A 6% increase in the quantity demanded of good X, when a 10% increase in the price of good Y i.e. option a is correct.

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The complete question is

At the current prices of goods X and Y, the quantity demanded of good X is 10 units, and the quantity demanded of good Y is 5 units. The cross-price elasticity of demand between goods X and Y is 0.6. A 10% increase in the price of good Y will result in which of the following?

a. A 6% increase in the quantity demanded of good X.

b. A 10% increase in the quantity demanded of good X.

Impact on the existing set of beliefs and values is small and the extent of the change to be introduced is relatively minor.

What is resistance?

The obstruction to current flow in an electrical circuit is measured by resistance. The Greek letter omega (Ω) represents the unit of measurement for resistance, known as ohms. Georg Simon Ohm (1784–1854), a German physicist who investigated the connection between voltage, current, and resistance, is the name given to the unit of resistance.

The movement of electrons is referred to as electric current. Consider the scenario of a person navigating a busy market and encountering difficulty moving from one shop to another to better understand resistance.

Hence,  the lowest level of resistance to change and the highest probability of successful change will more likely occur when the, impact on the existing set of beliefs and values is small and the extent of the change to be introduced is relatively minor.

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The probability of randomly selecting a male is dependent upon the gender distribution of the population from which the selection is made.

For example, if the population is evenly split between males and females, then the probability of randomly selecting a male would be 50%. Alternatively, if the population had more females than males, then the probability of selecting a male would be lower than 50%.

In general, the probability of randomly selecting a male is determined by the ratio of males to females in the population. For example, if there were twice as many females as males, then the probability of randomly selecting a male would be 33.3%. Similarly, if there were four times as many females as males, then the probability of randomly selecting a male would be 25%.

These probabilities can also be applied to specific populations. For example, the United States Census Bureau estimates that there are 101.8 males for every 100 females in the US population. Therefore, the probability of randomly selecting a male from the US population is approximately 50.9%.

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In a case wherby Becky has a paper route and It takes her 50 minutes to deliver newspapers to her 40 customers  the unit rate in newspapers per minute is 4/5 customers per minutes.

The concept of raqte will be used, Rate can be described as the degree of something measured per unit of something else.

From the question,

50 minutes  will be used to deliver newspapers to her 40 customers, then thye rate can be written as 40 customers/ 50 minutes

This can be simplified as :

4 customers/5 minutes

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The new profit per unit will be $7.

What are the aggregate prices?

The total amount of money required to create a product, provide a service, or carry out a project is referred to as the aggregate cost. The aggregate cost is also known as "total cost" because it includes all costs associated with the entire project or production.

We have,

aggregate prices + $3,

old profit of $4,

new profit?

Aggregate cost = total amount of money

new profit = old profit + increased price

new profit = 4 + 3

= 7

new profit = $7.

Hence, the new profit per unit will be $7.

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The correct option: 3. If AB BA for all square matrices B, then A must be a multiple of the identity matrix.

A square matrix called an identity matrix is one in which all of the primary diagonal members are one and all other components are zero.

As per the stated question-

Thus, if A is a multiple of the identity matrix if AB BA for all square matrices B.

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Answer:

y=ax+b

a=(y2-y1) /(x2-x1) =(-4+2)/(-4+5)=-2

The equation becomes:

y=-2x+b

Substitute (-2,-5) in the equation:

-5=-2(-2)+b

b=-9

y=-2x-9

The number terms in the given expression is 3, they are 5xy, -3x and -82.

The given expression is 5xy-3x-82.

An expression is a combination of terms that are combined by using mathematical operations such as subtraction, addition, multiplication, and division.

The terms involved in an expression in math are:

Constant: A constant is a fixed numerical value.

Variable: A variable is a symbol that doesn't have a fixed value.

In the given expression 5xy-3x-82, there are three terms, which are 5xy, -3x and -82.

Therefore, the number terms in the given expression is 3, they are 5xy, -3x and -82.

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Because both poems highlight the wonders of nature and the cyclical rhythm of the seasons, they are similar.

What is similarities?

The state of likeness can be referred to as a similarity. When two elements, things, ideas, or people resemble one another, they are said to be similar. Example: In terms of the number of sides, a square and a rectangle are comparable. The state of being different can be used to define a difference.

The moods and tones of two poems by the same author may be similar or dissimilar. You may, for instance, contrast and compare two poems written by the same author, such as Robert Frost's "A Prayer in Spring" and "A Late Walk." Because both poems highlight the wonders of nature and the cyclical rhythm of the seasons, they are similar. On the other hand, "A Prayer in Spring" has a joyful, lovely tone and a calm, appreciative atmosphere. On the other hand, "A Late Walk" has a gloomy, solemn tone and a downcast, pessimistic mood. Frost successfully contrasts the beauty of spring and the aridity of fall in each instance.

Look for similarities and variations in the two poems' language, style, and format. The length of each line or stanza, the meter, the rhythm, and the author's word choice should all be considered. For instance, some poets, like Dr. Seuss, favor monosyllabic words and short lines, while others, like William Wordsworth, favor polysyllabic words and lengthy lines or stanzas. Determine the level of technicality your teacher expects from your comparisons and contrasts, such as whether she wants you to talk about iambic pentameter, stressed syllables, and feet..

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If X1, X2, X3, and X4 are (pairwise) uncorrelated random variables, each having mean 0 and variance 1, the correlations of X1 + 2X2 and X2 + 3X3, the value will be 0.5.

Variance is the anticipated squared variation of a random variable from its population mean or sample mean in probability theory and statistics. Variance is a measure of dispersion, or how far apart from the mean a group of data are from one another.

Given,

X1, X2, X3, and X4 are (pairwise) uncorrelated random variables, each having mean 0 and variance 1,

Cov(X1+X2  , X2+X3)

= Cov(X1,X2) + Cov(X1,X3)+Cov(X2,X2)+Cov(X2,X3)

= 0+0+V(X2)+0

= 1

V(X1+X2)

=V(X1)+V(X2)+2Cov(X1,X2)

= 1+1-2*0

=2

Similarly V( X2+ X3) =2

So, correlation between ( X1+X2  and  X2+x3) is

Corr( X1+X2, X2+x3)

= Cov(X1+X2  , X2+X3) / \sqrt{}(V(X1+X2)*V( X2+ X3))

=1/\sqrt{}(22)

= 0.5

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Proved that at least one zero lies in the interval x  = 1 and x = 2 for the given polynomial

The polynomial is given by

f(x) = [tex]x^3[/tex] - 6x + 1

We have to show that at least one zero lies in the given interval for the given polynomial using the intermediate value theorem

When the value of x = 2

Substitute the value of x in the polynomial

f(2) = (2)^3 - 6(2) + 1

= 8 - 12 + 1

= -3

When the value of x = 3

Substitute the value of x in the polynomial

f(3) = (3)^3 - 6(3) + 1

= 27 - 18 + 1

= 10

Here the value of f(2) is negative and the value of f(3)  is positive therefore there is at least one real zero between x = 1and x = 2

Hence, proved that at least one zero lies in the interval x  = 1 and x = 2 for the given polynomial

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The answer is 1.79.

Division is breaking a number up into an equal number of parts.

Given that, 5.37 divided by 3

5.37÷3 = 1.79

Hence, 5.37 divided by 3 equals to 1.79.

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To explain further for each of the scenarios on how to solve the problem and show your work (set up the numbers in the equation) is given under.

How many two-apple combos can you create when you have three apples?

We may use the following combination equation to solve this problem: C(n, k) is equal to n! / (k! * (n-k)!. In this instance, the number of apples is n=3, and k=2  apples in each combination). With these values entered into the formula, we obtain:

C(3, 2) = 3! / (2! * (3-2)!) = 3 / 2 = 1.5

Given that a combination is a collection of things chosen without respect to their order, the solution to this puzzle is 1.5. However, because you cannot eat half an apple, the real number of combinations of two apples that may be formed is one.

How many possible combinations of two novels can you create if you have four?

C(n, k)=n!/(k!*(n-k))!

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An example of a linear function table is y=3x

A linear function is used to relate an independent variable and a dependent variable by introducing a rate of change(function) of which the independent variable is equated to find the dependent variable. Linear functions are those whose graph is a straight line. The following is the form of a linear function. a + bx = y = f(x). One independent variable and one dependent variable make up a linear function. X and Y are the independent and dependent variables, respectively.

for example:

y=3x hence is the dependent variable as it depends on the change of x and 3 is the rate of change which is a constant and is determined by the division of the values of y/x hence to make a table of y over x of which y is determined by the rate of change the unrelated variable x, 3.

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We use the quadratic formula to find the roots of the quadratic equation.

The quadratic equation is given below:

f(x) = a[tex]x^{2}[/tex] + bx + c = 0 where a, b, c

The system of equations used to find roots of equation is y = 12[tex]x^{3}[/tex]-5x and y=2[tex]x^{2}[/tex]+x+6.The given equation will be,12[tex]x^{3}[/tex]-5x=2[tex]x^{2}[/tex] + x + 6.

Now we take the root of equation into two parts:

1. Left-hand side

2. Right-hand side

Equation no.1 : 12[tex]x^{3}[/tex]-5x=0

Equation no.2: 2[tex]x^{2}[/tex]+x+6=0

Now we contains system of equation 0 with y.

So, the equation will be:

12[tex]x^{3}[/tex]-5x=y

2[tex]x^{2}[/tex]+x+6=y

However,system of equations can be used to find the roots of the equation 12[tex]x^{3}[/tex]-5x=2[tex]x^{2}[/tex]+x+6 is:
y = 12[tex]x^{3}[/tex]-5x and y = 2[tex]x^{2}[/tex]+x+6

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The probability of a random sample of parts produced by this machine being exactly defective = 0.15

Step-by-step explanation:

Let the percentage of defective part be 12%

and the sample be out of 7 , 2 are defective.

P(X=x) = (e-λ λ x)/ x!   [ poisson distribution]

Since, 12%  defective parts is produced by one of the machines of the time.

μ= (12%×7)= 0.84

the number of defective product =2

Therefore ,

P(X=2) = 0.15

Thus the required probability becomes 0.15

In Statistics, Poisson distribution is one of the important topics. It is used for calculating the possibilities for an event with the average rate of value. Poisson distribution is a discrete probability distribution.

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[tex]\huge \boxed{\sf C}\\\\\\\displaystyle \sf Substituting\ x=0 \\\\3(0)-3y=12\\\\-3y=12\\\\Dividing\ each\ side\ by\ -3 \\\\\frac{-3y}{-3} =\frac{12}{-3} \\\\y=-4 \\\\\\Substituting\ y=0 \\\\3x-3(0)=12\\\\3x=12\\\\Dividing\ each\ side\ by\ 3 \\\\\frac{3x}{3} =\frac{12}{3} \\\\x=4[/tex]

The length and height of the box with a volume of 70 cubic feet to reduce construction costs is 8.572 and 3.81.

Given that,

The base of an open box must be built with the length of the base being four times more than the width of the base. If building the base costs two dollars per square foot and building the four sides costs one dollar per square foot.

We have to identify the dimensions of a box with a volume of 70 cubic feet to reduce construction costs.

We know that,

The volume =70 cubic feet

L=4w    ------>equation(1)

Volume= l×w×h

70=4w×w×h

h=70/4w²     ------>equation(2)

The construction cost is

C= 4(lw)+2(wh)+2(hl)

C=4(4w×w)+2(w×70/4w²)+2(4w×70/4w²)

C=4(4w²)+2(70/4w)+2(70/w)

C=16w²+70/2w+140/w

Now,

To determine the minimum cost,

dC/dw

16w-70/4w²-140/w²=0

16w³=1/w²(70/4+140)

16w³=1/w²(315/2)

16w³=315/2

w³=315/32

w=2.143

From equation(1)

L=4w=4×2.143=8.572

H=70/4w²=70/4(2.143)²=3.81

Therefore, The length and height of the box with a volume of 70 cubic feet to reduce construction costs is 8.572 and 3.81.

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A population proportion is estimated to be around 541.

We can use the formula for the minimal sample size required for an interval estimate of a population proportion to address the problem.

n = pq ((Z/2) / E)^2

We use the conventional p and q, which are 0.5 and 0.5, because p is not defined.

The reason for this is that, if we choose values for p and q that range from 0.1 to 0.9, we will see that this is the point at which pq reaches its greatest value.

The value of (Z/2) will then be determined. We may determine that (Z/2) = 2.326 with a 98% confidence level by looking at the Z-table. We now begin changing the values in the formula above.

n = (0.5)×(0.5) × (2.326/0.05)^2

n = 541.027

n ≅ 541.

So, population proportion is estimated to be around 541

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SSS is the correct response because two triangles are considered to be congruent when all three of their respective sides and all three of their corresponding angles have the same measure.

Triangles that really are precisely the same in size and shape are said to be congruent. The symbol is in the term congruent. Two triangles are said to have been congruent when the three sides as well as three angles of one triangle have the identical dimensions as the three sides as well as three angles of another triangle.

Write the names of the vertices as shown below, starting with triangle DAB and triangle BCD.

AB = CD

DA = BC

The sides BD are identical in both triangles, so,

BD = BD

Since the triangles DAB and BCD have the same three sides, we may infer that they are congruent using the SSS property of a congruent triangle.

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When the space is 63 feet², the rectangle is 7 ft wide and 9 feet long.

The measurement that indicates the size of a region on a plane or curved surface is called area. Surface area refers to the area of an open surface or the border of a three-dimensional object, whereas the area of a plane region or plane area refers to the area of a form or planar lamina. The "space encompassed inside the perimeter or the boundary" of the specified form is what is referred to as a shape's area. Area is the entire amount of space occupied by a flat (2-D) surface or an object's form. On a sheet of paper, draw a square using a pencil. It has two dimensions. The area of a shape on paper is the area that it occupies.

Here,

Let w be the width and l be the length of the rectangle,

The area of rectangle= 63 feet²

area=w*l

l=w+2

w(w+2)=63

w²+2w=63

w²+2w-63=0

w²+9w-7w-63=0

w(w+9)-7(w+9)=0

(w-7)(w+9)=0

w=-9, 7

width=7 feet

length= 9 feet

The width of rectangle is 7 feet and length of rectangle is 9 feet when area is 63 feet².

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If a polynomial's degree is even and its leading coefficient is positive, then f(x) → ∞ as x → ±∞

A polynomial is a mathematical statement that solely uses the operations of addition, subtraction, multiplication, and positive-integer powers of variables. It is made up of indeterminates (also known as variables) and coefficients. x2 4x + 7 is an illustration of a polynomial with a single indeterminate x. x3 + 2xyz2 yz + 1 is a three-indeterminate example.

Numerous branches of mathematics and science use polynomials. For instance, they are used to define polynomial functions, which show up in contexts ranging from basic chemistry and physics to economics and social science, and they are used in calculus and numerical analysis to approximate other functions. Polynomial equations are used to encode a wide range of problems, from simple word problems to complex scientific problems. In higher mathematics, polynomials are used to build algebraic varieties and polynomial rings, two fundamental ideas in algebra and algebraic geometry.

The endpoints of the graphed lines entered and exited the same side of the image in each of the four images above. Like every positive quadratic you've ever graphed, the ends of the graphs of functions with positive leading coefficients came in and left off the top of the image. Like every negative quadratic you've ever graphed, the ends of the graphs of functions with negative leading coefficients came in and left out the bottom of the image.

Every polynomial of even degree will exhibit these characteristics. You can recall the end behavior of every even-degree polynomial if you can remember the behavior for quadratics (or for parabolas).

Hence from the above explanation:-

If a polynomial's degree is even and its leading coefficient is positive, then f(x) → ∞ as x → ±∞

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The measure of the angle of elevation of the sun to the nearest degree is 35 degrees.

Given:

A 50 foot tall tree casts a 72 foot shadow.

Here tree height is opposite to angle of elevation and shadow is adjacent to angle of elevation.

Let E be the angle of elevation.

tan(E) = opposite side / adjacent side

= 50/72

= 25/36

= 0.694

E = tan^-1(0.694)

= 34.99 degrees

≈ 35 degrees.

Therefore The measure of the angle of elevation of the sun to the nearest degree is 35 degrees.

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