HW 7: TOPICS, VIDEO, AND READINGS

a.) 4.29 seconds b.) 93 meters

A fireworks mortar is launched straight upward from a pool deck platform 3 m off the ground at an initial velocity of 42 m/sec. The height of the mortar can be modded by: h(t) = -4.9t² + 42t + 3 where h(t) is the height in meters and t is the times in seconds after launch. a.) Determine the maximum height's time b.) Determine the maximum height

$10,781 is the minimum cost

A medical equipment industry manufactures x-ray machines. The unit cost C (cost in dollar to make each X-ray machine) depends on the number of machines made. If x machines are made, then the unit cost is given by the function: C(x) = 0.4x² - 192x + 33,821 What is the minimum unit cost?

t = 0.52, 8.23 seconds

A model rocket is launched with an intial upward velocity of 140 ft/s. The rocket's height h (in feet) after t seconds is given by the following: h = 140t -16t² Find all values of t for which the rockets height is 68 feet.

a.) x = 85, y = 170 ft b.) 14,450 ft²

A parking area is to be constructed adjacent to a road. The developer has purchased 340 ft of fencing. a.) Determine dimensions for the parking lot that would maximize the area. a.) Then find the maximized area.

9 feet

A patio is configured from a rectangle with two right triangles of equal sizes attached at the two ends. The length of the rectangle is 20 ft. The base of the right triangle is 3 ft less than the height of the triangle. If the total area of the patio is 348 ft ² determine the base & height of the triangular portions.

a.) 1.02 seconds b.) 105 meters c.) 5.65 seconds

A stone is thrown from a 100-m cliff at an initial speed of 20m/sec at an angle of 30 degrees from the horizontal. The height of the stone can be modeled by: h(t) = -4.9t² + 10t + 100 a.) determine the time at which the stone will be at its maximum height. b.) determine the maximum height. Round to the nearest meter. c.) determine the time at which the stone will hit the ground?

a.) -4.9t² + 24t + 1 b.) t = -(-24) ± √(-24)² -4(4.9)(19)/2(49) - t = 3.9, 1.0 c.) 24 ± √-188.4/9.8 *SOLUTIONS ARE NOT REAL NUMBERS*

A toy rocket is shot straight upward from a launch pad of 1m above the ground level with an initial velocity of 24 m/sec. S = -1/2gt² + v₀t + s₀ a.) Write a model to express the height of the rocket s (in meters) above ground level b.) find the time(s) at which the rocket is at a height of 20m. Round to 1 decimal place. c.) Find the time(s) at which the rocket is at a hiegh of 40 m.

a.) (-2, 5) b.) (-4, 7) c.) function 1

Comparing properties of quadratic functions given in different forms. Function 1: (-6, 15) (-4, 7) (-2, 5) (0, 7) (1, 15) Function 2: f(x) = 2x² + 16x + 39 a.) find the vertex of function 1 b.) find the vertex of function 2 c.) which function has the smaller value?

a.) A(x) = -2x² + 280x b.) 70 meters c.) 9800 square meters

Deshaun has 280 meters of fencing. He will use it to form three sides of a rectangle garden. The fourth side will be along a house and will not need fencing. a.) find a function that gives the area A(x) of the garden in square meters in terms of x b.) Find the side length x that gives the maximum area c.) Find the maximum area

a.) two x - intercepts b.) no x-intercepts

Determine how many x-intercepts they have a.) 3x² + 12x + 5 = 0 b.) -x² + 4x - 5 = 0

a.) (-∞, 5/2) ∪ (5/2, ∞) b.) (-∞, ∞) c.) (-∞, 2] d.) (-∞, ∞)

Determine the domain and range of these functions: a.) f(x) = x + 3/2x - 5 b.) g(x) = x/x² + 4 c.) h(t) = √2 - t d.) |4 + a|

a.) f(x) = -2(x + 4) ² + 4 b.) f(x) = -3(x - 4)² + 5 c.) f(x) = -2(x + 5)² - 3

Find the equation of the quadratic functions: a.) vertex: (-4, 4) point: (-5, 2) b.) vertex: (4, 5) point: (5, 2) c.) vertex: (-5, -3) point: (-7, -11)

a.) minimum b.) x = 5 c.) -2

Find the maximum/minimum of a quadratic function: f(x) = 2x² - 20x + 48 a.) does the function have a minimum/maximum value? b.) Where does the min/max value occur? c.) What is the function's min/max value?

a.) (2x - 3)² < 0 { } b.) (2x - 3)² ≤ 0 {3/2} c.) (2x - 3)² > 0 (-∞, 3/2) ∪ (3/2, ∞) d.) (2x - 3)² ≥ 0 (-∞, ∞)

Find the solution set of these inequalities: a.) 4x² - 12x + 9 < 0 b.) 4x² - 12x + 9 ≤ 0 c.) 4x² - 12x + 9 > 0 d.) 4x² - 12x + 9 ≥ 0

(-∞, -5/3) ∪ (2, ∞)

Find the solution set of: (write in interval notation) 3x(x - 1) > 10 - 2x

a.) vertex: (3, 4) minimum b.) vertex: (-5, -1) maximum c.) vertex: (2, -3 ) minimum

Find the vertex of these functions and determine if its a maximum or minimum: a.) f(x) = 2x² - 12x + 22 b.) g(x) = -x² - 10x - 26 c.) g(x) = 3x² - 12x + 9

a.) x- intercepts: -6 vertex: (-6, 0) b.) x-intercepts: 2 vertex: (2, 0) c.) x - intercepts: -1 vertex: (-1, 0) d.) x - intercepts: -5, 1 vertex: (-2, -9) e.) x - intercepts: -2 -6 vertex: (-4, 4)

Find the x-intercepts and the vertex of a parabola (quadratic equation) a.) y= x² + 12x + 36 b.) y = -(x² - 4x + 4) c.) y = -x² - 2x - 1 d.) y = x² + 4x - 5 e.) y = -x² - 8x - 12

a.) upward b.) (0,0) c.) none d.) (0,0) e.) x = 0

Finding the vertex, intercepts, and axis of symmetry from the graph of a parabola: Use the graph of the parabola to fill in the table: a.) Parabola Opens: b.) Vertex: c.) Intersects the x-axis d.) intersects the y-axis e.) axis of symmetry

V = πr³-5πr²

Formula for the volume V of a cylinder with radius r and height h is as follows: V = πr²h Suppose the height of the cylinder is 5 units shorter than the radius. Rewrite V in terms of r only. h = r - 5 Substitute r - 5 for h in the formula for volume.

a.) one x-intercepts b.) two x - intercepts c.) no intercept

Given a quadratic function defined by f(x) = ax² +bx + c, • If b²-4ac = 0 • If b²-4ac > 0 • If b²-4ac < 0

a.) f(x) = 3(x +2)² - 7 b.) (-2, -7) c.) x = -0.47, -3.53 d.) (0, 5) e.) x = -2 f.) -7 g.) domain: (-∞, ∞) range: [-7, ∞)

Given f(x) = 3x² + 12x + 5 a.) write it in a vertex form b.) identify the vertex c.) x-intercepts d.) y-intercepts e.) axis of symmetry f.) minimum value g.) domain and range

a.) downward b.) (1, -3) c.) none d.) (0, -4) e.) x = 1 f.) -3 g.) domain: (-∞, ∞) range: (-∞, -3]

Given g(x) = -x² + 2x - 4 a.) Determine whether the graph of the parabola opens upward or downward b.) Identify the vertex c.) Determine the x-intercepts d.) Determine the y-intercepts e.) axis of symmetry f.) maximum value g.) domain and range

a.) downward b.) (-3/2, 59/4) c.) (0.717, 0) (-3.717, 0) d.) (0, 8) e.) x = 3/2 f.) 59/4 g.) domain (-∞, ∞) range: (-∞, 59/4)

Given: d(x) = -3x² - 9x + 8 a.) determine whether the graph of the parabola opens upward or downward b.) Identify the vertex c.) x-intercepts d.) y-intercepts e.) axis of symmetry f.) maximum value g.) domain and range

a.) downward b.) (1, 8) c.) (3, 0) (-1, 0) d.) (0, 6) e.) x = 1 f.) 8 g.) domain: (-∞, ∞) range: (-∞, 8]

Given: f(x) = -2(x - 1)² + 8 a.) determine if it opens upward/downward b.) identify the vertex c.) determine the x-intercepts d.) determine the y-intercepts e.) determine the axis of symmetry f.) determine the maximum/minimum value g.) write the domain and range in interval notation

a.) downward b.) (2, -1) c.) none d.) (0, -5) e.) x = 2 f.) -1 g.) domain: (-∞, ∞) range: (-∞, -1]

Given: f(x) = -x² + 4x - 5 a.) upward or downward b.) determine vertex c.) determine x-intercepts d.) determine y-intercepts e.) axis of symmetry f.) maximum/minimum value g.) domain and range

a.) (2, 0) (-2, 0) b.) (0, -4)

Given: f(x) = x² - 4 a.) find the x-intercepts b.) find the y-intercepts

The sheet aluminum should be folded 3 inches from the edges

Homeowner makes the trough from a rectangular piece of aluminum that is 20 in long and 12 in wide. Makes a fold along the two long sides a distance of x inches from the edges If he wants the trough to hold 360 in³ of water, how far from the edges should he make the fold? Note: V=LWH

a.) s = -16t² + 16t b.) t = 0.5

NBA basketball legend Micheal Jordan had a 48-in vertical leap. Suppose that Micheal jumps from ground level with an initial velocity of 16 ft/sec. a.) write a model to express Micheal's height above ground level t seconds after leaving the ground: s = -1/2gt² + vt₀ + s₀ g = 32 ft/sec² b.) Use the model from part a to determine how long it would take micheal to reach its maximum height of 48 in.

a.) upward b.) (-1, -8) c.) x = -1, -5 d.) (0, -6) e.) x = -1 f.) point (-1, -8) ; value = -8 g.) domain: (-∞, ∞) range: [-8, ∞)

Parabola Function: h(x) = 2(x + 1)² - 8 a.) determine whether the graph of the parabola opens upward or downward. b.) Identify the vertex c.) determine the x- intercepts d.) determine the y-intercepts e.) axis of symmetry f.) minimum/maximum value g.) domain & range

a.) A = 1/2h² - 2h b.) 4h³π c.) 8w

Rewrite a multivate function as a univariate function given a relationship between the variables: a.) A = 1/2bh b = h - 4 b.) V = πr²h r = 2h c.) P = 2L + 2w L = 3w

a.) t = 0.28, 0.72 b.) t = 1.97

Solve a quadratic equation with an irrational roots: a.) h = 1 + 5t - 5t² h = -2 b.) 0 = 70 - 4t - 16t²

a.) 90 meters; 16200 square meters b.) 110 meters; 12,100 square meters

Solve equations optimizing area by using a quadratic function: a.) 360 meters / A(x) = zx / A = 2x + z b.) 440 meters / A(x) = zx / A = 2x + 2z

zeros: 4/3, -1

Solve the inequality (find its x-intercepts): *you can use substitution* 3w² + w < 2(w + 2)

a. - x = 4 & x = -2 - x = 4 closed dot connecting to closed x = -2 b. - x = 3 & x = -5 - x = 3 closed dot pointing to the right direction - x = -5 closed dot pointing to the left direction c. - x = 2 & x = 6 - x = 2 open dot going in the the left direction - x = 6 open dot going in the right direction d. - x = 0 & x = -6 - x = - 6 open dot connecting to open dot x = 0

Solve the quadratic inequalities: a.) x² - 2x ≤ 8 b.) -x² - 2x ≤ -15 c.) x² - 8x > -12 d.) x² < -6x

a.) x = 40 width, y = 80 length b.) 3200 m²

Suppose that a family wants to fence in an area of their yard for a garden. One side is already fenced from the neighbor's property. (A = xy) a.) If the family has enough money to buy 160 ft of fencing, what dimensions would produce the maximum area for the garden? b.) What is the maximum area?

s = -16t + 48t + 1.5

Suppose that an object has an initial vertical position fo s₀ and initial velocity v₀ straight upward. The vertical position s of the object is given by: S = -1/2gt² + v₀t + s₀ g = accleration due to gravity (On Earth g = 32ft/sec²) t= time to travel v₀ = initial velocity s₀ = initial vertical position s = vertical position of the object at time t Suppose that a child tosses a ball straight upward from a height of 1.5ft with an initial velocity of 48 ft/sec.

length: 5.5 meters width: 12 meters

The area of a rectangle is 66m², and the length of the rectangle is 1m more than twice the width. Find the dimensions of the rectangle

vertex form

This equation represents: f(x) = a(x - h)² + k

quadratic function

This equation represents: f(x) = ax² + bx + c`

vertex form

This equation represents: f(x) = a(x - h)² + k

maximum value

This parabola has a MINIMUM/MAXIMUM value

minimum value

This parabola has a MINIMUM/MAXIMUM value

Hypotenuse of 5ft

Using Pythagorean Theorem: A window is in the shape of a rectangle with an adjacent right triangle is 2 ft less than the length of the hypotenuse. The length of the other leg is 1 feet less than the length of the hypotenuse. Find the length of the sides and then the hypotenuse.