John and Monica are paid ​$41.25 for their work. John worked 2.5 ​hours, and Monica worked 3 hours. They split the money according to the amount of time each of them worked. How much is John​'s share of the​ money?

Step-by-step explanation:

For solving this, it's important to know basic trigonometric functions:

sin = opposite side / hypotenuse

cos = adjacent side / hypotenuse

tan = opposite side / adjacent side

It's also important to know that it is necessary to know their values for special right triangles and angles of 30°, 45° and 60°

1. We are given angle if 45° and its adjacent side of 13. We want to know the opposite side (x) and hypotenuse (y). So:

tan = opposite side / adjacent side

tan 45 = x / 13

1 = x / 13

x = 13

Remember that tangent of 45° angle is 1.

We found the opposite side to be also 13.

To find y we can use:

sin = opposite side / hypotenuse

sin 45 = 13 / y

(√2)/2 = 13 / y

y = 13√2

Important to remember, sin 45 and cos 45 are (√2)/2

2. We are given 45° angle and hypotenuse of 30. We need to find opposite side (x) and adjacent side (y).

sin = opposite side / hypotenuse

sin 45 = x / 30

(√2)/2 = x / 30

x = 15√2

Also:

cos = adjacent side / hypotenuse

cos 45 = y / 30

(√2)/2 = y / 30

y = 15√2

We can conclude that in the right triangle, when angle is 45°, opposite and adjacent sides are equal.

3. We are given angle of 60° and adjacent side of 3. We need to find opposite side (y) and hypotenuse (x).

cos = adjacent side / hypotenuse

cos 60° = 3 / x

1/2 = 3 / x

x = 6

Note that cos 60° equals 1/2.

sin = opposite side / hypotenuse

sin 60° = y / 6

(√3)/2 = y / 6

y = 3√3

Note that sin 60° equals (√3)/2.

4. We are given angle of 30° and hypotenuse of 34. We need to find opposite side (y) and adjacent side (x).

sin = opposite side / hypotenuse

sin 30° = y / 34

1/2 = y / 34

y = 17

Note that sin 30° equals to cos 60° equals 1/2.

cos = adjacent side / hypotenuse

cos 30° = x / 34

(√3)/2 = x / 34

x = 17√3

Remember that cos 30° equals to sin 60° which is (√3)/2.

5. We are given an angle of 45° and hypotenuse of 10√2. We need to find the opposite side (y) and adjacent side (x).

sin = opposite side / hypotenuse

sin 45° = y / 10√2

(√2)/2 = y / 10√2

y = 10

To save time, we already said that opposite and adjacent sides are equal in right triangle with 45° angle, so x = y = 10.

6. We are given angle of 60° and opposite side of 25√3. We need to find adjacent side (y) and hypotenuse (x).

sin = opposite side / hypotenuse

sin 60° = 25√3 / x

(√3)/2 = 25√3 / x

x = 50.

tan = opposite side / adjacent side

tan 60° = 25√3 / y

√3 = 25√3 / y

y = 25.

Remember that tangent of 60° angle is √3.

7. We are given angle of 45° and hypotenuse of 2√14. We need to find adjacent side (x) and opposite side (y).

cos = adjacent side / hypotenuse

cos 45° = x / 2√14

(√2)/2 = x / 2√14

x = √28

Again, adjacent and opposite sides are equal in 45° right triangle, so x = y = √28.

8. We are given an angle of 30° and an adjacent side of 24. We need to find the opposite side (y) and hypotenuse (x).

tan = opposite side / adjacent side

tan 30° = y / 24

(√3)/3 = y / 24

y = 8√3

sin = opposite side / hypotenuse

sin 30° = 8√3 / x

1/2 = 8√3 / x

x = 16√3

Note that tan = sin/cos, so tan 30° = sin 30° / cos 30°

tan 30° = 1/2 / (√3)/2 = (√3)/3

I'm showing you this, so you don't have to memorize tan for special angles, you can find it from sin and cos.

9. We are given an angle of 60° and hypotenuse of 22√3. We need to find the opposite side (y) and adjacent side (x).

sin = opposite side / hypotenuse

sin 60° = y / 22√3

(√3)/2 = y / 22√3

y = 33

cos = adjacent side / hypotenuse

cos 60° = x / 22√3

1/2 = x / 22√3

x = 11√3

10. We are given an angle of 30° and the opposite side of √6. We need to find the adjacent side (x) and hypotenuse (y).

sin = opposite side / hypotenuse

sin 30° = √6 / y

1/2 = √6 / y

y = 2√6

tan = opposite side / adjacent side

tan 30° = √6 / x

(√3)/3 = √6 / x

x = √18

11. We are given an angle of 45° and an adjacent side of √10. We need to find the opposite side (x) and hypotenuse (y).

In this triangle opposite and adjacent sides are equal (45° angle), so x = √10

sin = opposite side / hypotenuse

sin 45° = √10 / y

(√2)/2 = √10 / y

y = √20 = 2√5

12. We are given an angle of 60° and the opposite side of 4√21. We need to find the adjacent side (x) and hypotenuse (y).

sin = opposite side / hypotenuse

sin 60° = 4√21 / y

(√3)/2 = 4√21 / y

y = 8√7

tan = opposite side / adjacent side

tan 60° = 4√21 / x

√3 = 4√21 / x

x = 4√7

13. Now things get a little trickier. We are given an angle of 30° and the opposite side of 17. We need to find the adjacent side (x).

Note that, at the moment, we are only dealing with this smaller triangle.

tan = opposite side / adjacent side

tan 30° = 17 / x

(√3)/3 = 17 / x

x = 17√3

Now, note that hypotenuse of the smaller triangle is the same length as the side z (adjacent and opposite side of 45° angle)

With Pythagorean theorem, we can find the hypotenuse of the smaller triangle, which equals to z.

z = √(17^2 + (17√3)^2)

z = 34

And now, finally to find y (hypotenuse of bigger triangle). We are given 45° angle and adjacent side z (34).

cos = adjacent side / hypotenuse

cos 45° = 34 / y

(√2)/2 = 34 / y

y = 34√2

14. Photo added

Correct responses;

Solutions:

1. An acute angle of the right triangle is 45°, therefore, the triangle is an isosceles triangle, and the leg lengths are equal.

Therefore;

The length of the hypotenuse side of an isosceles right triangle = A leg length × √2

Therefore;

2. An interior angle of the triangle = 45°

Therefore;

x = y

30 = x·√2

Which gives;

[tex]x = \dfrac{30}{\sqrt{2} } = \dfrac{30 \cdot \sqrt{2} }{2} = \mathbf{15 \cdot \sqrt{2}} = y[/tex]

3. The interior angle adjacent to the leg of length 3 = 60°

Therefore;

The hypotenuse side, x = 2 × adjacent leg length

Which gives;

In a right triangle having an interior angle of 60°, we have;

2 × Opposite leg length = √3 × The length of the hypotenuse

Therefore;

2 × y = √3 × x

Which gives;

2 × y = √3 × 6

4. The leg lengths are, x and y

An interior angle opposite to the leg length y is 30°

The hypotenuse side = 34

The length of the hypotenuse side = 2 × The leg length opposite the 30° angle

Therefore;

34 = 2 × y

[tex]y = \dfrac{34}{2} = \mathbf{17}[/tex]

The leg with length x is adjacent to the 30° angle, which gives;

2·x = √3 × 34

5. An acute interior angle is 45°

Therefore, the relationship are;

x = y

x·√2 = 10·√2

Which gives;

6. An acute interior angle is 60°, which in relation to the position of the sides gives;

x·√3 = 2 × 25·√3

Therefore;

x = 2 × 25 = 50

x = 50

y = x ÷ 2

Therefore;

y = 50 ÷ 2 = 25

7. An acute interior angle is 45°, which gives;

x·√2 = 2·√14

[tex]x = \dfrac{2 \cdot \sqrt{14} }{\sqrt{2} } = \sqrt{2} \times \sqrt{14} = \sqrt{28} = 2 \cdot \sqrt{7} [/tex]

8. An interior angle is 30°

With regards to the location of the variables, we have;

2 × 24 = x·√3

Therefore;

2·y = x

Therefore;

[tex]y = \dfrac{x}{2} [/tex]

Which gives;

[tex]y = \dfrac{16 \cdot \sqrt{2} }{2} = \mathbf{8 \cdot \sqrt{2} }[/tex]

9. An interior angle is 60°, which gives;

[tex]y = \dfrac{22 \cdot \sqrt{3} }{2} = 11 \cdot \sqrt{3} [/tex]

√3 × 22·√3 = 2 × x

x = 11·√3 × √3 = 33

10. An angle of the right triangle is 30°

With respect to the location of the variables, we have;

y = 2 × √6 = 2·√6

y·√3 = 2 × x

Therefore;

2·√6 × √3 = 2 × x

x = √(18) = 3·√2

11. Right triangle with a 45° interior angle

y = √(10) × √2 = √(20) = 2·√5

12. Interior angle of the right triangle is 60°

y·√3 = 2 × 4·√(21) = 8 × √7 × √3

[tex]x = \dfrac{y}{2} [/tex]

Therefore;

[tex]x = \dfrac{8 \cdot \sqrt{7} }{2} = \mathbf{ 4 \cdot \sqrt{7} }[/tex]

13. The ratio of the adjacent leg to the opposite leg to the 30° angle is √3, therefore;

The hypotenuse side = 2 × 17 = 34

The hypotenuse side of the right triangle having a 30° angle is a leg in

the right triangle that has the 45° angle, therefore;

14. In the 60° right triangle, we have;

x·√3 = 2 × 27 = 54 = 18 × 3

[tex]Length \ of \ the \ common \ side = \dfrac{x}{2} [/tex]

Which gives;

[tex]Length \ of \ the \ common \ side = \dfrac{18 \cdot \sqrt{3} }{2} = 9 \cdot \sqrt{3} [/tex]

Length of the common side = 9·√3

In the 30° right triangle, we have;

x·√3 = 9·√3

Therefore;

y = 2·x

Therefore;

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Answer:the number of adults that attended the game is 135.

the number of children that attended the game is 31

Step-by-step explanation:

Let x represent the number of adults that attended the game.

Let y represent the number of children that attended the game.

There were 166 paid admissions to a game. This means that

x + y = 166

The price was $2 for adults and $.75 for children. The amount of data taken in was $293.25. This means that

2x + 0.75y = 293.25 - - - - - - - - - - 1

Substituting x = 166 - y into equation 1, it becomes

2(166 - y) + 0.75y = 293.25

332 - 2y + 0.75y = 293.25

- 2y + 0.75y = 293.25 - 332

- 1.25y = - 38.75

y = - 38.75/- 1.25

y = 31

x = 166 - y = x = 166 - 31

x = 135

Answer:

Integrate information

Step-by-step explanation:

Answer:

1080 orders is made after 4hours

When profit exceeds $1000, the time will be 7:22 PM

Step-by-step explanation:

1.  Let No of order be O and time be T

O ∝ T

O = KT

K = O ÷ T

K = 18 ÷ 4 = 4.5

Total no of orders in 4hrs (240min) will be

O = 4.5 × 240 = 1080 orders

2. Total orders made from profit of $1000 is evaluated.

10 orders give $11

x orders will give 909.09091 orders

time required to reach 909.09091 orders is then evaluated

1080 orders is placed in 4hrs

909.09091 orders will be placed in x hours

upon cross multiplication

x = 3.3670034 hours which is equivalent to 202.020202 minutes

When that's added to 4:00 PM, the answer is 7:22PM minutes to the nearest minute.

1080 pies are sold in 4 hours based on their order rate, and the phone order profit will exceed $1,000 around 7:22 PM, assuming a constant profit rate.

The student asked two questions related to pizza orders.

Firstly, they wanted to know how many pies are sold in 4 hours based on a constant rate of 18 orders in a 4-minute period.

Secondly, they asked when phone order profit would exceed $1,000 assuming a profit of $11 on 10 orders.

Calculating Pies Sold in 4 Hours

To calculate the number of pies sold in 4 hours (240 minutes) when 18 orders are received every 4 minutes, we need to determine how many 4-minute periods are in 240 minutes.

This is 240 minutes / 4 minutes = 60 periods.

Since 18 orders are received per period, the total number of pies sold is 18 orders × 60 periods = 1080 pies.

Calculating When Phone Order Profit Exceeds $1,000

To calculate when the profit exceeds $1,000, note that for every 10 orders, a profit of $11 is made.

First, calculate the total number of orders needed to exceed $1,000 profit.

Given: $1,000 / $11 = approximately 91 sets of 10 orders.

Therefore, total orders needed = 91 sets × 10

                                                    = 910 orders.

Since 18 orders are taken every 4 minutes, we find the total time by dividing 910 orders by the rate of 4.5 orders per minute (18 orders / 4 minutes).

This gives approximately 202.2 minutes from the start time (4:00 PM), which rounds to 7:22 PM when the profit exceeds $1,000.

Missing Portion of actual question:

In actual question, the scale provided on the map is 1 inch = 2.5 miles. (Picture is attached)

Answer:

3 inch

Step-by-step explanation:

Part a):

Actual distance between Saugerties and Kingston is 10 miles (4*2.5) because in map distance is 4 inch while 1 inch is equivalent to 2.5 miles.

Part b):

Actual distance between Saugerties and Catskill is 15 miles hence on this map distance will be 6 inch (15/2.5) because 1 inch is equivalent to 2.5 miles.

Part c):

On another map where distance between Saugerties and Kingston is 2 inches (half of the one shown in the given map), the distance between Saugerties and Catskill will be 3 inches. As the map is scaled double than the map given in the question i.e. 1 inch = 5 miles.

Answer:

Step-by-step explanation:

Set your compass to length say AB and draw a circle centre A, without adjusting the compass, draw another circle with centre B. Name their intersectin C and D, erase the other part of the circle and leave just the intersect C and D. Draw a circle with centre A that passes through the intersection C and D, also draw a straight line that passes through centre C and D. Make an intersection say E on the circle by puting your comass on C and D. Join C and D to this intersection E.

Answer:

Racket must have cost 135.

Step-by-step explanation:

Given:

Percentile amount of shipping and handling charge = 7%

Actual amount of shipping and handling charge= 9.45

we need to find the Cost of racket.

Solution:

Let the cost of racket be 'x'.

Now we can say that:

Percentile amount of shipping and handling charge multiplied by cost of racket and then divided by 100 is equal to actual amount of shipping and handling charge.

framing in equation form we get;

 [tex]\frac{7}{100}x=9.45[/tex]

Multiplying both side by  [tex]\frac{100}{7}[/tex] we get;

[tex]\frac{7}{100}x\times \frac{100}{7}=9.45\times \frac{100}{7}\\\\x=135[/tex]

hence Racket must have cost 135.

Ari and his 3 brothers do not have enough money to take a school trip to Washington, D.C. they are in short of $39.

The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.

Given that, Ari and his 3 brothers want to take a school trip to Washington, D.C. They have $320 saved.

Ari and his 3 brothers

In total 4 of them saving $18 per week and they saved for 4 weeks

So, total money they saved

= 4×4×18

= $288

Total money they have =288+320

= $608

Difference in money

= 647-608

= $39

Hence, Ari and his 3 brothers do not have enough money to take a school trip to Washington, D.C. they are in short of $39.

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Final answer:

The depth of water around the boat, d(t), varies based on the tidal patterns. Tides are influenced by various factors including the moon's gravity, ocean depth, and local geography. Accurate prediction requires tidal charts and local data.

Explanation:

The depth of water around the boat, represented as d(t), varies based on time due to the tidal patterns. The tides are influenced by numerous factors such as the alignment and gravitational pull of the moon and sun, the depth of the ocean, and local geographical features like bays and estuaries. In the case of the Bay of Fundy, the tides can have an exceptionally large range due to its shape and the resonance of the tidal forces.

The frequency and height of tides will determine how often and to what extent the water level rises and falls, thus affecting the depth at any given hour. To forecast these tidal levels and ensure safe boating conditions, experts create tidal charts with high accuracy, taking into account local bathymetry and global tidal patterns. However, the actual mathematical representation of d(t) would require access to these local tidal patterns and data from the mentioned charts or measurement systems to create a function that accurately represents the changing water depth around the boat.

Answer:

Perimeter of the rectangle=6x+8 square units

Step-by-step explanation:

Given that area of rectangle is [tex]2x^2+7x+3[/tex]

Area of rectangle=lw square units

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

[tex]=x(2x+1)+3(2x+1)[/tex]

[tex]=(x+3)(2x+1)[/tex]

[tex]2x^2+7x+3=(x+3)(2x+1)[/tex]

Comparing the above equation with the given area we get

lw=(x+3)(2x+1)

Therefore length=x+3 and width=2x+1

To find the perimeter :

Perimeter of the rectangle=2(l+w) square units

[tex]=2((x+3)+(2x+1))[/tex]

[tex]=2(x+3+2x+1)[/tex]

[tex]=2(3x+4)[/tex]

[tex]=6x+8[/tex]

Therefore perimeter of the rectangle=6x+8 square units

The rate at which the surface area of the sphere is increasing can be found using related rates in calculus. By differentiating the volume formula with respect to time, we relate the rates of change for the radius and surface area, and then plug in the given volume increase rate when the radius is 3m.

The problem requires the application of related rates in calculus to find the rate at which the surface area of a sphere is increasing. To calculate this, we can use the formula for the volume of a sphere, which is V = \frac{4}{3}\pi r^3, and the formula for the surface area, which is S = 4\pi r^2. Given a rate of volume increase, \frac{dV}{dt} = 10 m^3/sec, we can differentiate the volume with respect to time to find the relationship between the rates of change of the radius and volume. Then we use the rate of change of the radius to find the rate of change of the surface area.

Step 1: Differentiate the volume with respect to time.

\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}

Step 2: Solve for \frac{dr}{dt} when the radius is 3m.

Step 3: Use the value of \frac{dr}{dt} to find \frac{dS}{dt} (the rate of surface area increase).

Step 4: \frac{dS}{dt} = 8\pi r \frac{dr}{dt}

Answer:

Ann had $296 money at first.

Step-by-step explanation:

Let the money her sister have be 'x'.

Given:

Ann had $198 more than her sister.

So we can say that;

Money Ann had = [tex]x+198[/tex]

Now Given:

mother gave Ann $20 and her sister $60.

Now,

Money Ann had after mother gave her money = [tex]x+198+20 = x+218[/tex]

Money her sister had after mother gave her money = [tex]x+60[/tex]

Also given:

Ann had twice as much money as her sister.

So we can say that Money Ann had after mother gave her money is equal to 2 times Money her sister had after mother gave her money.

framing in equation form we get;

[tex]x+218=2(x+60)[/tex]

Applying Distributive property we get;

[tex]x+218=2x+120[/tex]

Combining like terms we get;

[tex]2x-x=218-120\\\\x = \$98[/tex]

Money sister had first = $98

Money Ann had first = [tex]x+198 = 98+198 = \$296[/tex]

Hence Ann had $296 money at first.

Answer:  2. All college students

Step-by-step explanation:

A population is a large group of individuals that have some common feature by the researcher's point of interest.

Herero , the Department of Education wishes to estimate the proportion of all college students who have a job off-campus.

So , he need data of all students to compute the exact proportion.

But instead of that he surveyed 1600 randomly selected students which determine the sample.

Sample just gives the estimate of the parameter.

Hence, the population of interest to the Department of education is " All college students" .

The population of interest to the Department of Education, which wants to estimate the proportion of all college students who have a job off-campus, is all college students, not just the ones surveyed or who have off-campus jobs.

In the context of this question asked by the Department of Education, the population of interest refers to the group about which the Department wants to draw conclusions. The Department is looking for a proportion, specifically the proportion of all college students who have a job off-campus. Therefore, the population of interest to the Department of Education is option 2: all college students. This is because the Department wants to estimate a characteristic (having a job off-campus) of this entire group, not just of the students in the sample or of students who have off-campus jobs.

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Answer:

For this particular case they are interested on the amount of weight gained by randomly selecting some students, we need to remember that the weight can't be a discrete random variable since this random variable can take values on a specified interval and with decimals, so for this case the best conclusion is that we have  a continuous data set.

Step-by-step explanation:

Previous concepts

We need to remember that continuous random variable mans that the values are specified over an interval in the domain, so is possible to have decimal values for the possible outcomes of the random variable.

By the other hand a discrete random variable only can take integers for the possible outcomes of the random variable over the specified domain.

Solution to the problem

For this particular case they are interested on the amount of weight gained by randomly selecting some students, we need to remember that the weight can't be a discrete random variable since this random variable can take values on a specified interval and with decimals, so for this case the best conclusion is that we have  a continuous data set.

The data from a study on weight gains by college students in their first year is a continuous data set. This is because the data (weight gain) can take on any value including decimal values within a certain range.

The data collected from a study on weight gains by college students in their freshman year is considered to be a "continuous data set." This is because the weight gain, which is typically measured in pounds or kilograms, can take on any value within a certain range, including decimal values, reflecting the continuous nature of the data. For example, one student might gain 1.5 pounds, another might gain 2.3 pounds, and so on. It doesn't have to be in whole numbers, unlike in a discrete data set.

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Answer:

  the object is at 288 feet 2 seconds and 9 seconds after launch

Step-by-step explanation:

The equation of motion is given as ...

  h(t) = -16t² +176t

and we are asked to find when the object is at height 288 ft. Putting that number into the equation, we have ...

  288 = -16t² +176t

  -18 = t² -11t . . . . . . . . divide by -16

  -18 +30.25 = t² -11t +30.25 . . . . . . add (11/2)² to complete the square

  12.25 = (t -5.5)² . . . . . . simplify a bit

  ±3.5 = t -5.5 . . . . . . . . .square root

  t = 5.5 ± 3.5 = {2, 9}

The object is 288 feet high after 2 seconds or 9 seconds.

Answer:the park is located 0.808 miles away from the school

Step-by-step explanation:

Jonas is jogging from the Park to the school. The total number of miles that Jonas has jogged so far is 0.424.

He has 0.384 mile left to jog. This means that the distance of the park from the school would be the sum of the distance that he has jogged so far and the distance that he has left to jog. It becomes

0.424 + 0.384 = 0.808 miles

Answer:the speed of the wind is 20 miles per hour.

Step-by-step explanation:

A Plane flies at x miles per hour in still air. This means that the normal speed of the plane is x miles per hour.

Let y represent the speed of the wind.

Flying with a tailwind, it's speed is 485 miles per hour. This means that

the total speed of the plane would be x + y. Therefore

x + y = 485 - - - - - - - - 1

Against the wind, it's air speed is only 445 miles per hour. This means that

the total speed of the plane would be x - y. Therefore

x - y = 445 - - - - - - - - - 2

Adding equation 1 and equation 2, it becomes

2x = 930

x = 930/2 = 465

Substituting x = 465 into equation 1, it becomes

465 + y = 485

y = 485 - 465

y = 20 miles per hour

Answer:

D. all of the above

Step-by-step explanation:

In a programming software, depending on the type of input or coding given, the software application has the power to executes the following statement:

when the coding/input state x = y + 5

the computer will

A. retrieves the value of y (the value of y will have been stated earlier)

B. calculates the total of 5 and the value of y (sum the value of y with 5)

C. stores the result of the calculation in the variable x  (store the result of value of x in case if it is been call for again, or store nit until a new value is gotten for x)

them, the computer can either print out the result of xor wait for the next action.

the correct option for this question is D. all of the above

Answer:

Angle bisector

Step-by-step explanation:

median isn't applicable in this case as the roads from the streets are inclined at an angle.

altitude refers to height which is also not applicable

The perpendicular bisector is the locust of points equidistant from two points,

in this question the street are not seen as points but as lines which forms an angle and the bisection of this angle forms a locus where she can park her car. If she parks her car anywhere on the angular bisector of the two streets, she would be at equal distance from both streets.

Answer:

6.6%

Step-by-step explanation:

We find the cost of the bond. Yield to maturity is the yield for bond holder but cost for the  issuer like Easy Slider.

Formula is

P=CP(1-(1+x)^-n)/x + FV/(1+x)^n

where P is the price of bond in the market. So, the selling price of Easy Slider Inc bond is 928

CP= coupon payment. Here, CP is 10% of 1000. So, $100

FV= Face value. Here, FV is $1000

n= maturity of the bond. Here, n=15

x= cost of the bond before tax

putting the value in the equation

928=100(1-(1+x)^-15)/x + 1000/(1+x)^15

solving for x, we get 0.1100

Now, if we find out after tax then

0.1100(1-T)= After tax cost

0.1100(1-0.4)

0.066 or 6.66%

Answer:

  11,664 in³

Step-by-step explanation:

For a square side length of x, the girth is 4x and the length of the parcel is then allowed to be up to 108-4x.

The total volume is the product of the edge lengths, so is ...

  V = (108 -4x)(x²) = -4x³ +108x²

This will be a maximum where its derivative is zero:

  dV/dx = -12x² +216x = 0 = -12x(x -18)

This is zero for x=0 and for x=18.

The maximum volume parcel is 18 in by 18 in by 36 in, and has a volume of ...

  (18 in)(18 in)(36 in) = 11,664 in³

To find the largest possible volume of a rectangular parcel with two square sides adhering to postal regulations, we use a derived formula V = x²(108 - 4x) and apply optimization to find the maximum volume.

To answer the question about the largest possible volume of a rectangular parcel with two square sides that can be sent by fourth-class mail, we use the given postal regulation that states the girth plus the length of a parcel may not exceed 108 inches. Assuming we have a box with dimensions x (for the length and width of the square sides) and y (for the length), the girth is calculated as 4x (since we have two square sides) and the total allowed measurement is 4x + y <= 108 inches. The volume V of the parcel is x²y.

First, we express y in terms of x: y = 108 - 4x. We can then rewrite the volume formula in terms of x: V = x²(108 - 4x). To maximize the volume, we take the derivative of V with respect to x and set it to zero: dV/dx = 2x(108 - 4x) - 4x². Solving for x, we find the critical points which will give us the value of x that maximizes the volume. Plugging this x value back into V or y = 108 - 4x, we can find the maximum volume and the dimensions of the parcel. Without explicitly solving the calculus problem, the general approach is to use optimization via calculus or other mathematical techniques to maximize the volume function.

Answer:Kami sold 28 large figurines.

Step-by-step explanation:

Let x represent the number of large figurines that Kami sold.

Let y represent the number of small figurines that Kami sold.

This month kami sold 70 figurines in 2 sizes. This means that

x + y = 70 - - - - - - - - - - 1

The large figurines sold for $12 each and the small figurines sold for $8 each. The amount of money he received from the sales of the large figurines was equal to the amount of money he received from the sales of the small figurines. This means that

12x = 8y

y = 12x/8

y = 1.5x

Substituting y = 1.5x into equation 1, it becomes

x + 1.5x = 70

2.5x = 70

x = 70/2.5 = 28

y = 1.5x = 1.5 × 28 = 42

Answer:

The  height of the cone is 5.09 inches

Step-by-step explanation:

Given:

The volume of the cone  =  12 cubic inches

The diameter of the cone  = 3 inches

To Find :

The height of the cone  =  ?

Solution:

We know that the volume of the cone is

Volume  = [tex]\pi r^2\frac{h}{3}[/tex]

where

r is the radius of the cone

h is the  height of the cone

Now substituting the given values

[tex]12 = \pi (1.5)^2 \frac{ h }{3}[/tex]

[tex]\frac{12}{\pi 2.25} = \frac{h}{3}[/tex]

[tex]h = \frac{12}{ 2.25 \pi} \times 3[/tex]

[tex]h = \frac{ 12}{7.065} \times 3[/tex]

[tex]h = 1.698 \times 3[/tex]

h =  5.09  inches

Answer: OPTION A.

Step-by-step explanation:

By definition, the area of a triangle can be calculated with the following formula:

[tex]A=\frac{bh}{2}[/tex]

Where "A" is the area of the triangle, "b" is the base of the triangle and "h" is the height of the triangle.

In this case, you need to observe the triangular piece shown in the picture given in the exercise.

You can identify that the base and the height of the triangle are:

[tex]b=3.6\ ft\\\\h=4.8\ ft[/tex]

Therefore, you must substitute those values into the formula. Then:

[tex]A=\frac{(3.6\ ft)(4.8\ ft)}{2}[/tex]

Finally, you must evaluate in order to find the area of the triangle.

You get the following result:

[tex]A=\frac{17.28\ ft^2}{2}\\\\A=8.64\ ft^2[/tex]

Answer:20mph

Step-by-step explanation:

speed of plane = x mph

Speed of plane with wind = 485 mph

So therefore, speed of wind =

485 - x

Speed of plane against wind = 445 mph

So therefore, speed of wind =

x - 445

So therefore, equating the 2 equations of speed of wind

485 - x = x - 445

Collect like terms

485 + 445 = x + x

930 = 2x

2x = 930

x = 930/2 = 465 mph

Speed of wind = x = 465mph

So we put this value on any of the equations for speed of wind

Speed of wind = x - 445 = 465 - 445

= 20 mph

So therefore, speed of wind = 20mph

Answer:

The answer is 5300 units.

Step-by-step explanation:

The equation will be exactly lie below if the price is 5$:

[tex]5=-0.05x^2-0.3x + 8\\0=-0.05x^2-0.3x+3[/tex]

Roots of the parabol will be:

[tex]x_{1}=-11.3\\x_{2}=5.3[/tex]

In fact that there will be no negative production, The consumer surplus will be 5300 units

Answer:

The answer to your question is Antony bought 5 pens

Step-by-step explanation:

Data

pens = p = $0.35

pencils = n = $0.15

total amount = 12

total money spent = $2.80

Process

1.- Write equations

               p  +  n  = 12                -------------- (l)

        0.35p  +  0.15n = 2.80      --------------(ll)

2.- Solve the system of equations by elimination

Multiply equation l by - 0.35

         - 0.35p - 0.35n = -4.2

           0.35p + 0.15 n = 2.8

              0      - 0.2n = -1.4

Solve for n

                         n = -1.4 / -0.2

                         n = 7      He bought 7 pencils

3.- Find the value of p

              p + 7 = 12

              p = 12 - 7

              p = 5                He bought 5 pens

Answer:

[tex]H=-(t-3)^2+13[/tex]

Step-by-step explanation:

The path covered by the volleyball will be a downward parabola with the vertex being the highest point of the ball.

A general form of a downward parabola is given as:

[tex]y=-a(x-h)^2+k[/tex]

Where [tex](h,k)[/tex] is the vertex of the parabola and 'a' is a constant.

Now, let 'h' be the vertical height and 't' be the time taken.

So, the equation would be of the form:

[tex]H=-a(t-h)^2+k[/tex]

Now, as per question:

h = 2 seconds, k = 13 feet.

[tex]H=-a(t-3)^2+13[/tex]

Now, taking a = 1. So, the formula that can be used is:

[tex]H=-(t-3)^2+13[/tex]