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Express the area of a rectangular schoolyard in terms of the variable x, given 150 meters of fencing for three sides and the school wall as the fourth.
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A(x) = x * (150 - 2x)
Predict when two sloth populations, one experiencing exponential decay (0.85 factor) and the other linear decay (decrease by 2 units per year), will have equal populations.
Approximately 9 years.
Provide the exponential growth pattern for algae if it starts with 1/4 square meter on day 1 and doubles each day.
Day 1: 1/4, Day 2: 1/2, Day 3: 1, Day 4: 2, Day 5: 4, Day 6: 8
Calculate the volume of the open-top box when x is 3 cm using the function V(x).
V(3) = (10 - 2*3)² * 3 = 4² * 3 = 16 * 3 = 48 cm³
Describe the exponential decay pattern of a sloth population where each turn it's multiplied by 0.85.
The population is multiplied by 0.85 each year.
Describe the linear decay pattern of a second sloth population where 2 units are subtracted each year.
The population decreases by 2 units each year.
Express the area of a rectangular schoolyard as a function of width y if the total fencing for three sides is 150 meters and the fourth side is along a school wall.
A(y) = y * (150 - 2y)
Find the volume of an open-top box formed by cutting squares with side length x from the corners of a 10x10 cm cardboard.
V(x) = (10 - 2x)² * x
Calculate the area of the schoolyard when x is 40 meters using the function A(x).
A(40) = 40 * (150 - 2*40) = 40 * 70 = 2800 meters²
Formulate the recursive definition for a sequence P with initial value 5000, dividing by 10 each step.
P(1) = 5000, P(n) = P(n-1) / 10 for n ≥ 2
Write the equation for the volume of an open-top box created from a square piece of cardboard with side length 10 cm, with corner cutouts of length x.
Identify the maximum volume for an open-top box created by cutting squares with side length x from a 10x10 cm piece of cardboard, based on a graph.
Maximum volume ≈ 15 cubic inches when 0 ≤ x ≤ 2.5
Determine the domain for the function A(x) = x * (150 - 2x).
0 < x < 75
Define the sequence P(n) recursively given P(1) = 5000 and each subsequent term is one-tenth of the previous term.
Determine the appropriate domain for the volume function V(x) for an open-top box with cutout length x from a 10x10 cm cardboard.
0 < x < 5
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