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I NEED HELP PLEASE... HELPPPP MEEEE

Write a quadratic equation to represent each situation. Use your equation to answer each question. Be sure to define your variables and identify which is independent and which is dependent. Show all work. Round any decimals to the nearest hundredth.

1. A rocket is launched from a height of 10 meters with an upward velocity of 56 meters per second.

a. When will the rocket hit the ground?

b. What is the maximum height of the rocket?

c. After how many seconds does the rocket reach the maximum height?

2. A wooden deck of uniform width surrounds a rectangular pool as shown in the following image. The pool is 10 feet by 14 feet. What is the area of the wooden deck that surrounds the pool (not including the pool)?

a. If the deck is 3 feet wide, what is the area of the deck?

b. The area of the deck is 112 square feet. How wide is the deck?

3. A coffee shop sells 150 coffees a day at $3.50 apiece. They found for every $0.50 increase in price, they sell 10 fewer cups of coffee a day. The cups of coffee cost the shop $1.50 apiece.

a. How much should they charge to maximize their daily profits?

b. How many cups of coffee will they sell daily at that price?

c. What will their daily profits be?

User Sreekanth
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1 Answer

16 votes
16 votes

Answer:

56. In the Vector Addition Lab, Anna starts at the classroom door and walks:

2.0 meters, West

12.0 meters, North,

31.0 meters, West,

8.0 meters, South

3.0 meters, East

Using either a scaled diagram or a calculator, determine the magnitude and direction of Anna's resulting displacement.

Answer: 30.3 meters, 172 degrees

To insure the most accurate solution, this problem is best solved using a calculator and trigonometric principles. The first step is to determine the sum of all the horizontal (east-west) displacements and the sum of all the vertical (north-south) displacements.

Horizontal: 2.0 meters, West + 31.0 meters, West + 3.0 meters, East = 30.0 meters, West

Vertical: 12.0 meters, North + 8.0 meters, South = 4.0 meters, North

The series of five displacements is equivalent to two displacements of 30 meters, West and 4 meters, North. The resultant of these two displacements can be found using the Pythagorean theorem (for the magnitude) and the tangent function (for the direction). A non-scaled sketch is useful for visualizing the situation.

Applying the Pythagorean theorem leads to the magnitude of the resultant (R).

R2 = (30.0 m)2 + (4.0 m)2 = 916 m2

R = Sqrt(916 m2)

R = 30.3 meters

The angle theta in the diagram above can be found using the tangent function.

tangent(theta) = opposite/adjacent = (4.0 m) / (30.0 m)

tangent(theta) = 0.1333

theta = invtan(0.1333)

theta = 7.59 degrees

This angle theta is the angle between west and the resultant. Directions of vectors are expressed as the counterclockwise angle of rotation relative to east. So the direction is 7.59 degrees short of 180 degrees. That is, the direction is ~172 degrees.

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57. In a grocery store, a shopper walks 36.7 feet down an aisle. She then turns left and walks 17.0 feet straight ahead. Finally, she turns right and walks 8.2 feet to a final destination. (a) Determine the magnitude of the overall displacement. (b) Determine the direction of the displacement vector relative to the original line of motion.

Answer: (a) 48.0 feet; (b) 21 degrees from the original line of motion

This problem is best approached using a diagram of the physical situation. The three displacements are shown in the diagram below on the left. Since the three displacements could be done in any order without effecting the resulting displacement, these three legs of the trip are conveniently rearranged in the diagram below on the right.

Now it is obvious from the diagram on the right that the three displacement vectors are equivalent to two perpendicular displacement vectors of 44.9 feet and 17 feet. These two vectors can be added together and the resultant can be drawn from the starting location to the final location. A sketch is shown below.

Since these displacement vectors are at right angles to each other, the magnitude of the resultant can be determined using the Pythagorean theorem. The work is shown below.

R2 = (44.9 ft)2 + (17.0 ft)2 = 2305 ft2

R = Sqrt(2305 ft2)

R = 48.0 feet

The angle theta in the diagram above can be found using the tangent function.

tangent(theta) = opposite/adjacent = (17.0 ft) / (44.9 ft)

tangent(theta) = 0.3786

theta = invtan(0.3786)

theta = 20.7 degrees

This is the angle which the resultant makes with the original line of motion (the 36.7 ft displacement vector).

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Vector Addition || Vector Components || Vector Resolution

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58. A hiker hikes 12.4 km, south. The hiker then makes a turn towards the southeast and finishes at the final destination. The overall displacement of the two-legged trip is 19.7 km at 309 degrees . Determine the magnitude and direction of the second leg of the trip.

Answer: 12.7 km, 347 degrees

Like the previous problem (and most other problems in physics), this problem is best approached using a diagram. The first displacement is due South and the resulting displacement (at 309 degrees) is somewhere in the fourth quadrant. (It is in the fourth quadrant because 309 degrees lies between 270 degrees or due South and 360 degrees or due East.) For communication sake, we will refer to the first displacement as A and the second displacement as B. Note that A + B = R. Since the magnitude and direction of the resultant is known, the x- and y-components can be determined using trigonometric functions. Since the angle of 309 degrees is expressed as a counterclockwise angle of rotation with due East, it can be used as the Theta in the equation.

Rx = R•cos(theta) = 19.7 km • cos(309 deg) = 12.398 km

Ry = R•sin(theta) = 19.7 km • sin(309 deg) = -15.310 km (the "-" means South)

Whatever the magnitude and direction of B is, it must add on to vector A in order to give a southward displacement of 15.310 km and a eastward displacement of 12.398 km. This could be expressed by mathematical equations as

User NicklasF
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