Unit 11: Volume & Surface Area
Homework 1: Area of Plane Figures
Directions: Find the area of each figure. Round to the nearest hundredth where necessary.
2. and 11. all I need
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Answer:

Θ = 49.6°

Step-by-step explanation:

cos Θ = adj/hyp

cos Θ = 24/37

cos Θ = 0.64864

Θ = cos^-1 0.64864

Θ = 49.6°

a) The monthly payment for each loan individually is as follows:

                                 Loan A        Loan B         Loan C          Total

Monthly payments $133.82        $188.18      $195.00      $517.00

b) The total payments during the life of all three loans is $105,060, classified as follows:

                                 Loan A        Loan B         Loan C          Total

Total amount          $27,300     $49,680     $28,080    $105,060

c) The monthly payments after consolidating the three loans are $350.69.

d) The total payments over the 22 years period will be $92,581.47.

Note that student loans attract simple interest and not compound interest.

After the bank consolidates it, the interest system changes to compounding.

The total interest is computed by applying the annual percentage rate (APR) to the principal and the loan term.  The result is added to the principal to obtain the total payment for each loan.

The total payment is divided by the loan period to get the monthly payments.

But for the consolidated loan by the bank, we can use an online finance calculator to determine the monthly payments and total payment over the loan term as follows.

                                 Loan A        Loan B         Loan C          Total

Student loan            $12,000      $18,000       $13,500     $43,500

APR                             7.5%               8%              9%

Loan term                 17 years      22 years     12 years

Period in months        204              264           144

Total interests         $15,300      $31,680       $14,580      $61,560

Total amount          $27,300     $49,680     $28,080    $105,060

Monthly payments $133.82        $188.18      $195.00      $517.00

Total amount of loans = $43,500 ($12,000 + $18,000 + $13,500)

Bank APR = 8%

Loan term = 22 years

N (# of periods) = 264 months (22 years x 12)

I/Y (Interest per year) = 8%

PV (Present Value) = $43,500

FV (Future Value) = $0

Results:

PMT = $350.69

Sum of all periodic payments = $92,581.47

Total Interest = $49,081.47

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Answer:

1 1/6

Step-by-step explanation:

Hope it helps!

So in this question, we have to convert the rectangular coordinates to polar coordinates.

Polar coordinates are (r, Q).

To convert rectangular coordinates to polar coordinates we use that x is equal to  rcosQ and y is equal to rsinQ

In this question x is equal to -2 and y is equal to 5.

Hence we have -2 is equal to rcosQ and 5 is equal to rsinQ .

From here, we have tanQ equal to -2.5 . Hence, Q = tan⁻¹ (-2.5)

We have x²+y² =r² and also x²+y² = 5²+2²=29 .

Hence, r=√29 and Q = tan⁻¹ (-2.5) .

Therefore the polar coordinates of (-2,5) are ( √29, tan⁻¹ (-2.5)).

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The Sample of student at random from a large statistics class.

a) Sample space for Record the student's letter grade at the end of the course is , S = {A, B, C, D, F }. So, Correct option is option D.

b) Sample space for the student did or did not take a math class in each of the two previous years of school is S = {1st Year - Yes and 2nd Year - Yes, 1st Year - No and 2nd Year - Yes, 1st Year - Yes and 2nd Year - No, 1st Year - No and 2nd Year - No}. So, correct option is option C.

Sample Space: It is defined as the list of all possible outcome for given experiment.

a) record of Students grade are A, B, C, D, F.

These are all possible outcome for Student's letter grade at the end of course.

S= {A, B, C, D, F }

b) Possible outcome for whether student did or

did not take math class in each of the previous year of school is,

1st - Yes & 2nd - yes , 1st-yes & 2nd no

1st- NO & 2nd yes, 1st-NO & 2nd_NO

Hence, S = { 1st year - Yes & 2nd year - Yes,

1st year-NO & 2nd year- Yes, 1st year- yes & 2nd year - NO, 1st year - No & 2nd year - No}

Hence 3rd option is correct option.

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Complete question :

choose a student at random from a large statistics class. describe a sample space s for each of the following. (in some cases you may have some freedom in specifying s.)

a) Record the student's letter grade at the end of the course.

A) S = {0,10,20,30,...,100}

B) S = {0,1,2,...,100}

C) S = {Fail, Passed, Good, Very Good, Excellent}

D) S = {A, B, C, D, F}

b) Ask whether the student did or did not take a math class in each of the two previous years of school.

A) S = {Yes, No}

B)S = {Yes for each of the two years, No for each of the two years}

C) S = {1st Year - Yes and 2nd Year - Yes, 1st Year - No and 2nd Year - Yes, 1st Year - Yes and 2nd Year - No, 1st Year - No and 2nd Year - No}

D) S = {1st Year - Yes, 1st Year - No, 2nd Year Yes, 2nd Year - No}

Answer:

The ladder rests against the wall at 6ft.

The ladder in Figure 2 rests about 3ft higher than the ladder in Figure 1.

Step-by-step explanation:

We can assume both of the figures display the ladder, ground, and wall forming a right triangle because the assignment is titled "apply the Pythagorean theorem", suggesting we should use the Pythagorean theorem and both figures are right triangles.

The Pythagorean theorem states that the sum of each leg squared is equal to the hypotenuse squared, or a^2 + b^2 = c^2, where "a" and "b" are legs and "c" is the hypotenuse.

At what height does the ladder in Figure 1 rest on the wall?

First, identify which sides are the legs and which is the hypotenuse. The hypotenuse is always the longest side and is the side opposite from the right angle. Thus, the hypotenuse in Figure 1 is the ladder length, and the two legs are the distance the ladder is from the wall and the height of the ladder against the wall. In other words, c = 10, a = 8, and b = the missing side length. Now, plug in the values to the Pythagorean theorem:

8^2 + b^2 = 10^2

64 + b^2 = 100

b^2 = 36

b = 6

The height at which the ladder in Figure 1 rests on the wall is 6 ft.

Approximately how much higher up the wall does the ladder in Figure 2 rest compared to the ladder in Figure 1?

We can set up the equation: Figure 2 ladder height - Figure 1 ladder height. We already know how where the ladder height of Figure 1 is, so we only need to solve for the ladder height in Figure 2 to get the result. Applying the same method as in the previous problem, we can input the values of Figure 2 into the Pythagorean theorem:

4^2 + b^2 = 10^2

16 + b^2 = 100

b^2 = 84

b = sqrt84

b ~ 9 ft

Now, we can find the difference between the Figure 2 ladder height and Figure 1 ladder height to get the answer to the question:

9 ft - 6 ft = 3 ft

Answer:

A.) Yes, because 0.064 is less than 0.4

Step-by-step explanation:

The equation [tex]0.4^3 = 0.064[/tex] is a counterexample to the given conjecture because 0.064 is less than 0.4. This shows that the cube of a number is not always greater than the number.

In general, the cube of a number is the result of multiplying the number by itself three times. For example, the cube of 2 is [tex]2 * 2 * 2 = 8[/tex], and the cube of 5 is [tex]5 * 5 * 5 = 125[/tex]. The cube of a number is always greater than the number when the number is positive, because a positive number multiplied by itself three times is always a larger number. However, when the number is negative or a decimal, this is not always the case.

In the given equation, 0.4 is a decimal number, and [tex]0.4^3[/tex] is the cube of 0.4. The result of this calculation is 0.064, which is less than 0.4. This means that the cube of 0.4 is not greater than 0.4, and therefore the given conjecture is false.

Answer:

[tex]a_8=-\dfrac{729}{15625}[/tex]

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Geometric sequence}\\\\$a_n=ar^{n-1}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\\phantom{ww}$\bullet$ $r$ is the common ratio.\\\phantom{ww}$\bullet$ $a_n$ is the $n$th term.\\\phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]

Given geometric sequence:

[tex]\dfrac{5}{3},\;-1,\;\dfrac{3}{5},\;...[/tex]

To find the common ratio, divide a term by the previous term:

[tex]\implies r=\dfrac{a_3}{a_2}=\dfrac{\frac{3}{5}}{-1}=-\dfrac{3}{5}[/tex]

Substitute the found common ratio and given first term into the formula to create an equation for the nth term:

[tex]a_n=\dfrac{5}{3}\left(-\dfrac{3}{5}\right)^{n-1}[/tex]

To find the 8th term, substitute n = 8 into the equation:

[tex]\implies a_8=\dfrac{5}{3}\left(-\dfrac{3}{5}\right)^{8-1}[/tex]

[tex]\implies a_8=\dfrac{5}{3}\left(-\dfrac{3}{5}\right)^{7}[/tex]

[tex]\implies a_8=\dfrac{5}{3}\left(-\dfrac{2187}{78125}\right)[/tex]

[tex]\implies a_8=-\dfrac{729}{15625}[/tex]

Total 26, 250 passengers are flew in June in an airline flights between Chicago and Denver .

An Equations with roots , such as the square root cannot be solved until its root is eliminated. Once we have isolated the square root on one side, we can eliminate it by raising both sides of the equation to powers equal to the square root. For example, if the square root is only one-sided, we can square both sides. We have , An airline offers daily flights between Chicago and Denver. Cost function of flight is,

C = 0.2x + 1

where x represents the number of passengers (in thousands) in flight.

To calculate how many passengers flew each day in June, we need to set the function to 2.5 (since the cost is in millions of dollars) and solve for the variables. This means taking the square root equal to 2.5. Therefore, to solve for the variables contained in its square root, we need to square both sides. This process and solution are described below,

√(0.2x + 1) = 2.5

Squaring both sides

=> (√(0.2x + 1) )² = ( 2.5 )²

=> 0.2x + 1 = 6.25

=> 0.2 x = 6.25 - 1 = 5.25

=> x = 5.25/0.2

=> x = 26.25

Now, we know that our variable i.e x measures passengers in thousands, so total passenger flew in June are 26.25 × 1000 = 26, 250 .

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The product of the given values would be = 1,125. That is option A.

Multiplication is defined as one of the major arithmetic operation along with others such as addition, subtraction and division which is used to to determine the product of two or more values.

The rule of multiplication is that

Therefore;

= (-25)× (15) × (-3)

= -375 × -3

= 1,125.

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Answer:

Step-by-step explanation:

Evan bought 18 postcards in 9 days

since he got 12 in 6 days that means he got 2 each day so 9x2=18

or if you add up by 2 9 times you would get 18 :)

The smallest leg of the triangular garden is 5 feet while the biggest leg is 12 feet.

The three-sided shape known as a triangle is sometimes used to allude to it.

The area of a triangle is defined as (1/2) base height, and the sum of all three angles within a triangle will be 180°.

As the given triangular garden has been drawn below,

By Pythagoras' theorem,

x² + (x + 7)² = (x + 8)²

x² + x² + 49 + 14x = x² + 64 + 16x

x² - 2x - 15 = 0

x² - 5x + 3x - 15 = 0

x(x - 5) + 3(x - 5) = 0

(x + 3)(x - 5) = 0

x = -3,5

Since length can never be negative thus x = 5.

The smallest leg = 5 feet

Biggest leg = 5 + 7 = 12 feet

Hence "The triangular garden's smallest leg is 5 feet long, while its longest leg is 12 feet".

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Answer:

B. 8

Step-by-step explanation:

I’m glad to help you

The principal that would need to be invested is  $4202.

We know that the principal has to  do with the amount of money that you have to invest so as to be able to obtain an interest. The interest in this case is a simple interest and it is charged on the principal. The amount is the sum of the principal and the interest.

We now have;

A = I + P

A = amount

I = interest

P = principal

But I = PRT/100

R = rate

T = time

We have;

A = PRT/100 + P

5000 = P * 9.5 * 2/100 + P

5000 = 19P/100 + P

5000 = 19P + 100P/100

5000 * 100 = 119P

P = 500000/119

P = $4202

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Reagan will grow 1/2 inch than her brother if Reagan grew 9/10 of an inch and her brother grew 2/5 of an inch by using ratio and proportion concept,

When b does not equal 0, an ordered pair of numbers a and b, represented as a / b, is said to be a ratio. Two ratios are set to be equal in an equation called a proportion. For instance, if there is 1 boy and 3 girls, the ratio would be written as 1: 3 (there are 3 girls for every boy), meaning that there are 1 in 4 boys and 3 in 4 girls.

A: b a/b is the ratio formula, which may be used to calculate any two values. A:b::c:da:b::c:da, on the other hand, is how the percentage formula is written.

Given,

The height grown by Reagan=9/10 inch

The height grown by Reagan brother=2/5 inch

(9/10)-(2/5)

=1/2 inch

Therefore, Reagan will grow 1/2 inch than her brother.

Therefore,

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Based on the two (2) parallel lines m||n, the value of x is equal to 4.

The vertical angles theorem is also referred to as vertically opposite angles theorem and it states that two (2) opposite vertical angles that are formed whenever two (2) lines intersect each other are always congruent, which simply means being equal to each other.

In Geometry, when two (2) parallel lines are cut by a transversal, the pair of vertically opposite angles that are formed would always be equal to each other. In this context, we have the following;

m∠2 = m∠3

7x - 3 = x + 21

7x - x = 21 + 3

6x = 24

x = 24/6

x = 4.

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Answer:

15.

Step-by-step explanation:

The common ratio = 60/120 = 1/2

So the next term is 30 * 1/2 = 15.

The region bounded by the equations has a precise area of 21.33 units.

A boundary or some set of constraints are imposed on a bounded region. To put it another way, the size of a bounded shape cannot be infinite. Anything that is bound must be able to be contained within certain parameters.

x2 = 4y2 from equation (1), y3 = 4y2 y3 - 4y2 = 0 y2(y - 4) = 0 y2 = 0 or y-4 = 0 or y = 4 These will be our integration bounds. We have given that x2 = y3 --------- (1) and x = 2y --------- (2).

Therefore, we will express area as A = [- 4]04 = [- 4 - 0] = 64 - 85.33 A = -21.33, or A = 21.33. This indicates that the area of the bounded region is 21.33 units.

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The solution to the system of equations, y = 4x - 9 and y = x - 3, is: (2, -1)

The coordinates of the ordered pair that will make both equations true is the solution to the system.

Given the linear equations as:

y = 4x - 9 --> equation 1

y = x - 3 --> equation 2

Make both linear equations equal to each other and find the value of x. Thus:

4x - 9 = x - 3

Combine like terms

4x - x = 9 - 3

3x = 6

Divide both sides by 3

3x/3 = 6/3

x = 2

To find the value of y, substitute x = 2 into equation 1:

y = 4(2) - 9

y = 8 - 9

y = -1

The solution is, (2, -1).

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Answer:

801 boxes

Step-by-step explanation:

[tex]\frac{7209}{9}[/tex] = 801

The hyperbola's standard equation is [(x2/a2) - (y2/b2)] = 1, where X denotes the transverse axis and Y denotes the conjugate axis.

A hyperbola is a particular kind of smooth curve that lies in a plane and is classified by its geometric characteristics or by equations for which it is the solution set. A hyperbola is made up of two mirror images of one another that resemble two infinite bows. These two sections are known as connected components or branches. A hyperbola is a geometric shape in mathematics where the difference between the lengths from any point on the figure to two fixed locations is a constant (the Greek letter o literally means "overshooting" or "excess"). We refer to the two fixed spots as foci (plural of focus).

Given

The equation for the hyperbola

The hyperbola's standard equation is [(x2/a2) - (y2/b2)] = 1, where X denotes the transverse axis and Y denotes the conjugate axis.

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Answer: Lets do 2 1/7

Step-by-step explanation: first you will start off by turning it to an improper fraction.

you multiply the denominator with the whole number (2x7) which you get 14 then add the numrinator with the number you got from multiplying (14+1= 15 then you get 15/7

Then after that divide the fraction 15/7 and you get 2 with a remander of 1  

Here is a website you can use for fractions into whole numbers-https://www.cuemath.com

Answer:

72% are black haired - 5 students becomes 65% students are black haired.

meaning 7% are 5 students

Base = Percentage/Rate

Base = 5/0.07

Base = 71 students

71 (students) * 0.72 (rate of black haired) = 51

51 - 5 (leave the class) = 46 students

71 (students) * 0.65 (rest of black hair) = 46 students

Step-by-step explanation:

The Option that can be proven is:

Definitions, assertions, and methods are linked in a proper fashion to get the desired outcome in mathematical proof.

This procedure helps pupils understand the rationale behind the statement. This is also true of counterexamples and their important significance in mathematics.

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The interest rate, to the nearest hundredth of a percent, which would be required in order for Grace to end up with $190 is 2.38%.

Mathematically, compound interest can be calculated by using this formula:

A(t) = P(1 + r)^{t}

Where:

Substituting the given parameters into the compound interest formula, we have;

190 = 140(1 + r)^{13}

190/140 = (1 + r)^{13}

1.3571 = (1 + r)^{13}

Taking the 13th root of both sides of the equation, we have:

[tex]\sqrt[13]{1.3571} = \sqrt[13]{(1 + r )^{13} }[/tex]

1.0238 = 1 + r

r = 1.02378 - 1

Interest rate, r = 0.0238

Interest rate, r = 2.38%

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The function f is guaranteed by the extreme value theorem to have an absolute maximum and an absolute minimum on the closed interval a. [-5, 5].

The extreme value theorem states that if a function f is continuous on a closed interval [a, b], then it must have both an absolute maximum and an absolute minimum on that interval. Since the function f is given as being defined on the closed interval [-5, 5], it must have both an absolute maximum and an absolute minimum on this interval.

We can check that the function f is continuous on the interval [-5, 5] as follows:

First, we need to check that the function is defined at every point in the interval. The function f is defined as f(x) = x+4/(x-1)(x+3). The only points at which this function is not defined are x = 1 and x = -3, which are not within the interval [-5, 5]. Therefore, the function is defined at every point in the interval.

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The quotient will be 102.

What is a quotient?

In arithmetic, a quotient is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics and is commonly referred to as the integer part of a division, or as a fraction or a ratio.

The number 8976 is called the numerator or dividend, and the number 88 is called the denominator or divisor.

The quotient of 8976 and 88, the ratio of 8976 and 88, as well as the fraction of 8976 and 88 all, mean (almost) the same:

8976 divided by 88, often written as 8976/88.

= 8976/88

= 102

Hence, the quotient will be 102.

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Richard will increase the amount of time he studies each night by 25%.

The percentage is defined as a ratio expressed as a fraction of 100.

To find the percentage increase, we need to first find the difference between the two amounts of time.

In this case, the difference is 50 minutes - 40 minutes = 10 minutes.

Next, we need to divide the difference by the original amount of time and multiply by 100% to express the answer as a percentage.

In this case, the percentage increase is :

⇒ (10 / 40) x 100% = 25%.

Therefore, he will increase the amount of time he studies each night by 25%.

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