LINEAR EQUATIONS & INEQUALITIES
Moana is sailing her canoe. She starts 15 miles away from Motunui and sails at a constant rate of 8 miles per hour. Write and solve an equation to find how many hours, h, it will take her to reach an island 55 miles away.
h = 5 hours (Equation: 15 + 8h = 55)
At Tiana's Palace, a customer buys 3 orders of beignets and 2 bowls of gumbo for $21. Another customer buys 1 order of beignets and 4 bowls of gumbo for $27. Find the individual cost of an order of beignets (x) and a bowl of gumbo (y).
Beignets = $3, Gumbo = $6 (System: 3x + 2y = 21 and x + 4y = 27)
Experiment 626 accidentally enters a cloning machine. The number of Stitches triples every hour. If Jumba started with 4 Stitches, the population is modeled by P = 4(3)^t. How many Stitches will there be after 5 hours
972 Stitches (Solution: P = 4(3)5)
Tarzan swings from a tree. His height in feet above the ground is modeled by h(t) = -16t2 + 32t + 20, where t is time in seconds. Find the maximum height Tarzan reaches during his swing.
36 feet (Solution: Vertex is at t = 1 second; h(1) = 36)
The function f(c) = 4c + 15 models the total minutes Cinderella spends cleaning, where c represents the number of rooms she sweeps. Evaluate and interpret the meaning of f(6).
39 minutes; it takes her 39 minutes to clean 6 rooms. (Solution: f(6) = 4(6) + 15)
Scrooge McDuck has $5,000 in gold coins. Every day he goes swimming, he accidentally loses $15 worth of coins in the drains. He needs to keep at least $4,610 in the bin to remain the richest duck. Write an inequality to find the maximum number of days (d) he can swim.
d <= 26 days (Inequality: 5000 - 15d >= 4610)
Judy Hopps issues a total of 200 parking tickets on her first day. Some are $25 violations and others are $50 violations. If her total fines collect $6,500, how many of each ticket did she issue?
140 tickets at $25, 60 tickets at $50 (System: x + y = 200 and 25x + 50y = 6500)
The Powhatan village expands their farming land. The width of a cornfield is x + 3 and the length is 2x - 5. Expand the polynomial to find the expression representing the total area of the field.
2x2 + x - 15 (Solution: (x + 3)(2x - 5))
Rapunzel drops a heavy frying pan out of her tower window from a height of 144 feet. The equation tracking its height is h(t) = -16t2 + 144. How many seconds (t) will it take for the pan to hit the ground (h = 0)?
t = 3 seconds (Solution: -16t2 + 144 = 0)
The number of ghosts arriving at the ballroom party follows an arithmetic sequence: 99, 106, 113, 120... Find the common difference d and write the explicit formula an for the n-th term.
d = 7; an = 7n + 92 (Solution: an = 99 + (n - 1)7)
The royal guards print 120 wanted posters for Flynn Rider. Maximus the horse eats 4 posters every hour. Write an equation to determine how many hours (t) it takes for the number of posters to drop to exactly 32.
t = 22 hours (Equation: 120 - 4t = 32)
Joy and Sadness are sorting core memories. Together they sort 45 memories. Joy sorts 3 more than twice the number of memories that Sadness sorts. How many memories did each sort?
Joy = 31 memories, Sadness = 14 memories (System: J + S = 45 and J = 2S + 3)
The magical floor tiles in Encanto lay themselves out in a square grid with an area represented by x2 + 10x + 25. Factor the polynomial to find the expression representing the length of one side of the square room.
Side length = x + 5 (Solution: Factoring reveals (x + 5)2)
The track of the secret lab rollercoaster dips into a dark ravine following the path y = x2 - 8x + 12. Find the x-intercepts of this quadratic path to determine where the coaster enters and exits the ravine.
x = 2 and x = 6 (Solution: Factoring gives (x - 2)(x - 6) = 0)
Mulan fires a rocket to trigger an avalanche. The snow travels down the mountain according to the geometric sequence 3, 6, 12, 24... meters per second. Find the explicit formula an and calculate the speed at the 8th term.
an = 3(2)n-1; speed is 384 meters per second (Solution: 3 * 27)
Ralph earns 150 points for every brick building he smashes, but he loses 50 points every time Fix-It Felix repairs a window. In one round, Ralph smashes (b) buildings and Felix repairs 6 windows. If Ralph's final score is 2,100 points, find b.
b = 16 buildings (Equation: 150b - 50(6) = 2100)
Mr. Incredible (x) and Elastigirl (y) strike the Omnidroid. Together, they land 85 punches. If Elastigirl lands 15 punches fewer than Mr. Incredible, how many punches did each hero land?
Mr. Incredible = 50 punches, Elastigirl = 35 punches (System: x + y = 85 and y = x - 15)
The energy output of a child's laugh canister is modeled by the expression (4x3y2)3. Simplify the expression completely using the laws of exponents.
64x9y6
Merida shoots an arrow straight up into the air. The path is modeled by y = -5t2 + 40t, where y is height in meters and t is seconds. How many seconds pass before the arrow hits the ground?
t = 8 seconds (Solution: -5t(t - 8) = 0)
The Lost Boys are simplifying a recipe for flying potion. Simplify the rational expression completely: (x2 - 9) / (x2 + 5x + 6).
(x - 3) / (x + 2) (Solution: Factoring gives [(x-3)(x+3)] / [(x+2)(x+3)])
Baymax starts a medical mission with 100% battery. When walking, his battery drains at 2.5% per hour. When flying, it drains at 6% per hour. If he walks for 4 hours, write and solve an inequality to find the maximum hours (f) he can fly before his battery drops below 42%.
f < 8 hours (Inequality: 100 - 2.5(4) - 6f > 42)
Phil has Hercules lifting marble pillars (x) and catching flying Harpies (y). Pillars weigh 500 lbs each and Harpies weigh 40 lbs each. Hercules lifts/catches 12 items total, managing a total weight of 2,300 lbs. How many pillars did he lift?
4 pillars (and 8 Harpies) (System: x + y = 12 and 500x + 40y = 2300)
The storage box for Buzz's extra rocket fuel is a rectangular prism. The dimensions are x, x + 2, and x - 2. Write a simplified polynomial expression for the total volume of the box.
x3 - 4x (Solution: V = x(x + 2)(x - 2))
Maleficent shoots a ball of green fire that follows the trajectory h(x) = -x2 + 12x - 20. Solve for x using the quadratic formula to find the exact horizontal boundaries where the fireball is at ground level (h = 0).
x = 2 and x = 10 (Solution: Using quadratic formula on -x2 + 12x - 20 = 0)
Lightning McQueen drives one lap around Radiator Springs (d miles) at 120 mph and a second lap at 150 mph. The rational expression for his average speed over the two laps reduces to: 2 / ((1/120) + (1/150)). Simplify this complex fraction to find his exact average speed.
133.33 mph (or 133 and 1/3 mph)