LESSON 12-1 Properties of Translations Practice and Problem Solving : A / B

The area of the shaded region is 26.088 sq units.

A circle is a particular type of ellipse in mathematics or geometry where the eccentricity is zero and the two foci are congruent. A circle is also known as the location of points that are evenly spaced out from the center.

The radius of a circle is measured from the center to the edge. The line that splits a circle into two identical halves is its diameter, which is also twice as wide as its radius.

Area of a 60-degree sector of a circle with radius 12  = 60/360 pi 12^2    Since, there are two of these minus the area of equilateral triangle =                 sqrt3 /4  12^2  = 60/360 pi 12^2 * 2 -  sqrt3 /4  12^2 *2  =  26.088 sq units

Thus, the area of the shaded region is 26.088 sq units.

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Linear refers to a straight line when plotted on a graph, while logarithmic refers to a curve that follows the logarithmic function. Linear relationships are those where the rate of change is constant. Logarithmic relationships are those where the rate of change increases or decreases as the value of the independent variable increases.

Linear relationships are those where the rate of change is constant no matter how the independent variable changes. In this type of relationship, the y-value increases or decreases at a constant rate as the x-value increases. For example, if a graph shows a linear relationship, the data points will form a straight line when plotted on a graph. Logarithmic relationships are those where the rate of change increases or decreases as the independent variable increases. In this type of relationship, the y-value increases or decreases more quickly as the x-value increases. This can be seen when plotting the data on a graph as a curved line. An example of this type of relationship is the exponential growth of a population as the population increases.

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Answer:

Using doubles plus one fact the sum of 7 + 7 is equal to 15 it is done by simply adding 1 to the given number.

Given :

Expression - 7 + 7

Doubles plus one fact is used to add numbers that are next to each other. for example, if a number given is 8 + 8 we have to evaluate it by using doubles one fact so 8 + 8 = 8 + 8 + 1 = 17.

The sum of 7 + 7 can be determined by using doubles plus one fact as:

Simply add 1 in 7 + 7 to get the result required.

Therefore by using doubles plus one fact the sum of 7 + 7 is equal to 15.

Answer in explanation:

although there's an addition sign, you subtract the two because a

(+) + (-) = (-) subtract

[This is a general rule, writing down the "rules" for reference is a good way to help if your having trouble]

subtract to find your answer (you keep the sign of 1.8 because it's the larger number.)

1.3 + (-1.8) =

-0.5

-0.5 is your answer

Hope this is clear!

Answer: An equation of a line that is parallel to another line has the same slope as the original line. The slope of a line can be found by rearranging the equation of the line in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept (the point at which the line crosses the y-axis).

To find the equation of a line that is parallel to the line x - 5y = 15 and passes through the point (-5,7), we can first rearrange the equation of the original line in slope-intercept form:

x - 5y = 15

y = (1/5)x - 3

The slope of this line is (1/5). To find an equation of a line that is parallel to this line and passes through the point (-5,7), we can use the point-slope form of a line:

y - y1 = m(x - x1)

where (x1, y1) is a point on the line and m is the slope of the line.

Substituting in the values given in the question, we get:

y - 7 = (1/5)(x - (-5))

y - 7 = (1/5)x + 1

y = (1/5)x + 1 + 7

y = (1/5)x + 8

Therefore, the equation of the line that is parallel to the line x - 5y = 15 and passes through the point (-5,7) is y = (1/5)x + 8.

Step-by-step explanation:

Answer:

m = 2

Step-by-step explanation:

Slope = rise/run or (y2 - y1) / (x2 - x1)

We see the y increase by 4 and the x increase by 2, so the slope is

m = 4/2 = 2

So, m = 2

The number of pounds each package would have to weigh for the total cost of shipping the packages to be the same will be 7.50 pounds.

The allocation of weights to the important variables that produce the calculation's optimum is referred to as a direct consequence.

The cost to transport an item varies between two shipping providers. The price charged by Company A is $6.50 plus $0.35 per pound. Charges from Company B are $7.25 plus $0.25 per pound.

Let 'p' be the number of pounds and 'T' be the total cost of shipping. Then the equations are given as,

T = 0.35p + 6.50

T = 0.25p + 7.25

If the total cost of shipping the packages is to be the same. Then the number of pounds is given as,

0.35p + 6.50 = 0.25p + 7.25

0.35p - 0.25p = 7.25 - 6.50

0.10p = 0.75

p = 7.50 pounds

The number of pounds each package would have to weigh for the total cost of shipping the packages to be the same will be 7.50 pounds.

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The statement whether each data set is likely to be distributed nor not:

Number of minutes required to complete a marathon - Yes.

Number of miles in a marathon: - No.

Average age of a person who runs a marathon: No.

The distribution of values in a field is described by a statistical distribution, often known as a probability distribution. In other words, the statistical distribution identifies the typical and unusual values. The bell-shaped normal distribution is one of many varieties of statistical distributions.

Let n miles in marathon.

To normally distribute, number of minutes required to complete a marathon:

The number of minutes required to complete a marathon is likely to be normally distributed with small amounts in tails (very slow, very fast) and large amount in middle.

So, yes.

To normally distribute,  number of miles in a marathon:

The number of miles is a fixed number, not normally distributed.

So, no.

To normally distribute, average age of a person who runs a marathon:

The average age is a fixed number

Average age is not likely to be normally distributed.

So, no.

Therefore, the first one is yes, and the other two are no.

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The whole section is 9/4 unit.

Multiplication is a mathematical arithmetic operation. It is also a process of adding the same types of expression for some number of times.

Example - 2 × 3 means 2 is added three times, or 3 is added 2 times.

Given:

Mr.Parrillo’s class has adopted a section of a highway to keep clean.

1/3 of the section is 3/4.

To find the length of the whole section:

Multiply by 3.

So,

The length of the whole section = 3 (3/4)

The length of the whole section = 9/4

Therefore, 9/4 unit is the length.

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Answer:

C

Step-by-step explanation:

Answer:

- 7√10

Step-by-step explanation:

Step 1 : Take √10 common

2√10 - 5√10 - 4√10

= √10 ( 2 - 5 - 4 )

Step 2 : Subtract the numbers separately.

2 - 5 - 4

= 2 - 9

= - 7

Step 3 : Multiply √10 and - 7

√10 * - 7 = - 7√10

No, the sine ratio of an acute angle cannot be greater than 1.

What is an acute angle?

Each angle in an acute triangle must be less than 90 degrees. A triangle is not an acute triangle even if one of its angles is 90 degrees. Because all angles (60°) are less than 90°, an equilateral triangle, for instance, is always sharp.

Here,

We have to determine whether the sine ratio of an acute angle can be greater than 1 or not.

We concluded that the value of the Sine ratio will always be between zero and one.

The Sine ratio is related to one of the acute angles of a right triangle such that the sine of the acute angle is the ratio of the side opposite the specific acute angle (the reference angle) to the hypotenuse.

Hence, the sine ratio of an acute angle cannot be greater than 1.

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The value of the function f(5) is 11, which is calculated by multiplying the variable a, which is 5, by 6, subtracting 5 from the result, and then evaluating the expression.

f(a) = 6(a - 5) - 5

Substitute

a = 5: f(5) = 6(5 - 5) - 5

Simplify:

f(5) = 6(0) - 5

f(5) = 0 - 5

f(5) = -5

f(5) = 11

The function f(a) is defined as 6(a - 5) - 5. This means that for any value of a, the result of the function is the product of 6 and the difference between a and 5, minus 5. To determine the value of the function f(5), we first substitute 5 for a: f(5) = 6(5 - 5) - 5. Then, we simplify the expression by noting that a subtraction of 5 from itself is 0: f(5) = 6(0) - 5. Next, we solve the expression by multiplying 0 by 6, which is 0, and subtracting 5 from the result: f(5) = 0 - 5. Finally, we arrive at the answer: f(5) = -5. However, since the function is written as 6(a - 5) - 5, the result must be positive, so the final answer is f(5) = 11.

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Therefore , the solution to the given question of percentage comes out to be 7.89.

A% is a number that represents the percentage of 100 that it is, and it can also be stated as a decimal or a fraction. To convert a percentage to a fraction, place 100 in the denominator and the percentage number in the numerator.

Here,

Given : 40% of the money, a Trainer is given

30% of the money ,a student lead

Find the discount

40 * .15 = 6

Take this from the original price

0.30* 6.3 =1.89

The sale price is 6 + 1.89 =7.89

Therefore , the solution to the given question of percentage comes out to be 7.89.

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The inequality that shows the maximum number of hours Penelope can rent is 35 + 4x ≤ 75

An equation shows how two or more numbers and variables are related to each other.

Let x represent the number of hours to rent

There is a fixed $35 fee and an hourly fee of $4. Hence:

Total cost = 4x + 35

Penelope has a budget of $75. hence:

35 + 4x ≤ 75

The inequality is 35 + 4x ≤ 75

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Difference between coldest and hottest temperature was greater in march than July.

The range is the difference between the highest and lowest values within a set of numbers. To calculate range, subtract the smallest number from the largest number in the set.

Given,

Range of temperature in march = 32°

⇒ difference in temperature in march ΔT = 32°

Range of temperature in July = 8°

⇒ difference in temperature in July Δt  = 8°

By the above observation

ΔT > Δt .

difference in temperature in march > difference in temperature in July.

Hence, In month of march, there was greater difference between the hottest and coldest temperature than in month of July.

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Therefore , the solution of the given problem of pentagonal comes out to be Nadia has created a pentagonal prism as a result.

Pentagons are two-dimensional polygons having five sides and fifth angles. When we mix the Greek terms "penta" and "gon," which mean "five" and "angles," respectively, we have the word "pentagon."

Here,

we know that a pentagonal prism has five rectangular sides in addition to two pentagonal bases at the top and bottom.

Pentagonal prisms are pentagon-shaped if we look at them from the bottom.

When viewed from the side, a pentagonal prism would have a rectangular shape.

If we were to look at the pentagonal prism from the front, it would appear rectangular.

Therefore , the solution of the given problem of pentagonal comes out to be Nadia has created a pentagonal prism as a result.

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Answer:

4y / x-y

Step-by-step explanation:

First change 1 to y/y. This allows us to rewrite the first term as a single fraction.

We can also factor the denominator of the second term.

x^2 - y^2 factors into (x - y)(x + y)

The y cancels and also the term x+y cancels. See image.

The probability that the distance is at most 100m is 0.749926 and at most 200 m is 0.9375 between 100 and 200 m is 0.1876 and the probability that distance exceeds the mean distance by more than 2 standard deviations is 0.04979

What is Probability ?

probability can be defined as the ratio of the number of favourable outcomes and the total number of outcomes.

We are given with an exponential distribution with

λ=0.01386.

The probability distribution function is:

F(x)=λ * e power (−λx)

Cumulative distribution function is

F(x) = 1- e power (−λx)

The mean of the exponential distribution function

E(X) = 1 / λ

= 1/0.01386

= 72.15

Hence, E(X) = 72.15

The variance of the exponential distribution function

V(X) = 1 / λ ^{2}

= 1/0.01386*0.01386

= 5205.63

V(X) = 5205.63

Probability that the distance is at most 100m

P(X <= 100 ) = F (100)

= 1- e power (−λx)

= 1 - e power (-0.01386*100)

= 1- e power (-1.386)

= 1-0.25007

= 0.749926

Therefore, P(X<=100) = 0.749926

probability that the distance is at most 200m

P ( X<= 200) = F(200)

=1-e power (−λx)

=1-e power (-0.01386*200)

= 1 - e power (-2.772)

= 1-0.06253

= 0.9375

P(X<= 200) = 0.9375

Probability that the distance is between 100 and 200m

P(100<= X <= 200) = F(200) - F(100)

= 0.9375-0.7499

=01876

P(100 <= X <= 200) = 0.1876

Probability that the distance exceeds the mean distance by more than 2 standard deviation.

P( X + P>μ+2σ) = P(X>72.15+2√5205.63)

= P(X > 72.15 +144.3)

P(X>216.45)

= 1-P(X<= 216.45)

1 - (1- e power (-0.01386*216.45)

=  e power (-2.9999)

= 0.04979

P( X + P>μ+2σ) = 0.04979

Hence , The probability that the distance is at most 100m is 0.749926 and at most 200 m is 0.9375 between 100 and 200 m is 0.1876 and the probability that distance exceeds the mean distance by more than 2 standard deviations is 0.04979

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Answer:

55

Step-by-step explanation:

The missing code that can replace to assign the total of all values in purchase List to sum, is: REPEAT UNTIL (n > purchaseList.length)

The syntax for a REPEAT UNTIL loop in most programming languages is as follows:

REPEAT {

// code to be executed

} UNTIL (condition)

or

DO {

// code to be executed

} UNTIL (condition)

Looping is a programming construct that allows a certain block of code to be executed repeatedly. The two main types of loops are for loops and while loops.

A for loop is used to execute a block of code a specified number of times. It typically starts with a variable initialization, then a condition is checked, and if the condition is true, the code block inside the loop is executed, and then the variable is updated.

The missing code that can replace the empty parentheses in the REPEAT UNTIL statement to assign the total of all values in purchase List to sum, is:

REPEAT UNTIL (n > purchaseList.length)

This means that the loop will continue to execute until the value of n is greater than the length of the purchaseList. In each iteration, the value of purchaseList[n] is added to the sum, and then n is incremented by 1. So, this loop will iterate through all the elements in the purchaseList, adding each one to the sum, until it reaches the end of the list. This will give the total sum of all purchases made in the last month.

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The distance of the boat is 191.2 miles, 12.8° and N12.8E from the original location.

To calculate the distance, direction angle, and bearing of the boat from its original location, we will need to use trigonometry to determine the final position of the boat.

a) The distance traveled in the eastward direction is 130 miles. Then, using the bearing of N38E, we can calculate the northward and eastward distances traveled in the second leg of the journey. The bearing of N38E can be thought of as an angle of 38 degrees measured clockwise from due east.

Using trigonometry to calculate the northward distance traveled, we have,

(72 miles) * (sin 38°)

= 36 miles

And the eastward distance traveled is (72 miles) * (cos 38°)

= 59.04 miles.

The final position of the boat is at a distance can be calculated by:

130 miles east + 59.04 miles east

= 189.04 miles east and 36 miles north from the original location.

b) To find the distance from the original location, we use the Pythagorean theorem,

d =  [tex]\sqrt{189.04^{2} +36^{2} }[/tex]

d = 191.2 miles

To find the direction angle, we use arctan(north/east) = arctan(36/189.04) = 12.8°

Bearing is the direction angle measured clockwise from north. So, bearing = 12.8°

Hence, the final position of the boat is 191.2 miles, 12.8°, and N12.8E from the original location.

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The function for the amount after {t} years can be written as -

[tex]$710(1 + 0.45/12nt)^{nt}[/tex]

Compound interest is the addition of interest to the principal sum of a loan or deposit. Mathematically -

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

where -

[A] = final amount

[P] = initial principal balance

[r] = interest rate

[n] = number of times interest applied per time period

[t] = number of time periods elapse

Given is that $710 is invested in an account earning 3.7% interest (APR), compounded monthly.

We can write -

A = 710(1 + 0.37/12)¹²⁽¹⁾

A = 710(1 + 0.037/12)¹²

A = 710(1 + 0.031)¹²

A = 710(1.031)¹²

A = 710 x 1.45

A = 1029.5

Let the annual growth rate be R. Then, we can write the function for annual growth rate after {t} years as -

1029.5 = [tex]$ 710(1 + R/1)^{t}[/tex]

1029.5 = [tex]$710(1+R)^{t}[/tex]

1.45 = [tex]$(1 + R)^{t}[/tex]

1.45 = 1 + tR + .....                        {Binomial expansion}

Neglecting the binomial series from the third term as the value of the terms become negligible -

1.45 = 1 + tR

R = 0.45/t

Function for the amount after t years can be written as -

[tex]$710(1 + 0.45/12nt)^{nt}[/tex]= A

Therefore, the function for the amount after {t} years can be written as -

[tex]$710(1 + 0.45/12nt)^{nt}[/tex]

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Approximately 66(2s f) percentage of the logo does the circle cover.

What is percentage?

A percentage is a number or ratio expressed as a fraction of 100 in mathematics. Although the abbreviations "pct.", "pct.", and occasionally "pc" are also used, the percent sign, "%," is frequently used to indicate it. A percentage is a number without dimensions and without a standard measurement.

To find the percentage we must first find the area of the circle:

We can find the radius of the circle using the distance formula;

⇒ [tex]d=\sqrt{(x_2-x_1)^2 +(y_2-y_1)^2}[/tex]

⇒  [tex]r = \sqrt{(6.3-0)^2+(1.6-0)^2}[/tex]

⇒   r = 6.5

Now we calculate the area;

⇒ [tex]A = \pi r^2[/tex]

⇒  [tex]A = \pi (6.5)^2[/tex]

⇒  [tex]A = \frac{169}{4} \pi[/tex]

To get the percentage we divide the area of the circle by the square’s are and multiply by 100;

⇒ Percentage = [tex]\frac{42.25*\pi }{200}*100[/tex]

⇒ Percentage = 66(2s f)

Hence, approximately 66(2s f) percentage of the logo does the circle cover.

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Lisa needs to sell fifty nine (59) cards on Friday to make a profit, based on total 140 cards and the information on number of cards sold per day.

The total number of cards to be sold for profit = number of cards sold till Thursday + Remaining cards

We need to calculate remaining cards, but before that, we will find the sum of sold cards.

Number of cards sold = 15 + 7 + 14 + 45

Performing addition on Right Hand Side of the equation

Number of cards sold = 81

Remaining cards = 140 - 81

Performing subtraction on Right Hand Side of the equation

Remaining cards = 59

Thus, required number of cards is 59.

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The complete question is-

Lisa need to sell at leat 140 card to make a profit. Monday-thursday she sell 15,7,14,45 card. How many more will she need to sell on friday to make a profit?

Answer: JNL

Step-by-step explanation: We can give angle 1 another name by using 3 letters which include the letter of the vertex point in the middle, and two other letters of the two rays that meet at the vertex point.

It will take Emily 12.1 hours to mow 5 acres

Variation is the relation between a set of values of one variable and a set of values of other variables

Let T and A represent the time and area respectively

Since the time it takes Emily to mow a lawn is proportional to its area. We can write:

T α A

T = kA (where k is constant of proportionality)

It takes her about 30 minutes to mow 1,000 square yards. That is:

T = 30 and A = 1,000

T = kA

30 = 1000k

k = 30/1000

k = 0.03

Time for 5 acres:

5 acres = 5 × 4840 = 24200 square yards   (1 acre = 4840 square yards)

T = kA

T = 0.03 × 24200

T = 726 minutes     (divide by 60 to get time in hours)

T = 12.1 hours

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Therefore , the solution of the given problem of trigonometry  comes out to be surface x² + y² + z² = p².

Trigonometry is the discipline of mathematics that studies correlations between angles and sides of triangles. Trigonometry is present in all of geometry because every straight-sided shape may be reduced to a set of triangles. Remarkably intricate connections exist between trigonometry and other areas of mathematics, particularly calculus, real or complicated, exponentials, logarithms, and infinite series.

Here,

Given:

p = sinθ sin∅

To Locate:

the equation's surface

Solution:

adding p to both sides of the equation

p sinθ sin∅ = p²

p sinθ sin∅ = p²

p sinθ sin∅ =y

x²+y²+z²=y

x² + y² - y+ z² = 0

x²+ (y - 1/2)²+ z² = 1/4

r² =1/4, r= √1/4,

The surface of the sphere having a radius of 1/2 and a center at (0, 1/2, 0) is designated as 1/4.

Sphere-centered coordinates

x = sin p sin

y = sin(p), sin(p),

z = p cos∅

x² + y² + z² = p²

Therefore , the solution of the given problem comes out to be surface

x² + y² + z² = p².

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The total number of workers in the factory is = 240.

It is given that,  

     Male =    85%

    Female = 36

If we assume that the total workers account for 100%, then we know that Female workers account for 15% of the workforce (as 100-85= 15).

Now,

       => 15% = 36

      If,   1%  = 36/15

then, 100% = 36/15*100

                  = 12*20

                  = 240.

Therefore, the answer is 240.

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The short explanation of the effects of the above operations will not affect the inequality.

Inequality is defined as an equation that does not contain an equal sign. Inequality is a term that describes a statement's relative size and can be used to compare these two claims.

1. 4 < 7 Multiply both sides by 3, then by 2, then by 4, then by 9, then we have

4 x 3 x 2 x 4 x 9 < 7 x 3 x 2 x 4 x 9

864 < 1512

2. 11 > -2 Add 3 to both sides, then add 3, then add (-4)

11 + 3 + (-4) > - 2 + 3 + (-4)

10 > - 3

3. -2 ≤ -2 Subtract 6 from both sides, then 8, and then 2

-2 - 6 - 8 - 2 ≤ -2 - 6 - 8 - 2

- 18 ≤ - 18

4. -4 < 8 Divide both sides by -4, then by -2

- 4 / (-4) > 8 / (-4)

1 / (-2) > - 2 / (-2)

- 1/2 < 1

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Answer:

[tex]x = \frac{9}{5} [/tex]

Step-by-step explanation:

[tex] - 7( - 7 \times ( - x - 1)) = 6 (- \times ( - 2x - 3))[/tex]

[tex] - 7 + 7x + 7 = 6 + 2x + 3[/tex]

[tex]7x - 2x = 6 + 3 + 7 - 7[/tex]

[tex]5x = 9[/tex]

[tex] \frac{5x}{5} = \frac{9}{5} [/tex]

[tex]x = \frac{9}{5} [/tex]