0. Functions

Find the linear function whose graph passes through the point (3, 2) and is parallel to the line $y=3x+8$ .

Solve the equation sin 2Θ = 1, for 0 ≤ Θ < 2π .

Evaluate cos⁻¹(cos(5π/4)).

Solving trigonometric equations Solve the following equations.

tan x = 1

Solving trigonometric equations Solve the following equations.

cos²Θ = 1/2 , 0 ≤ Θ < 2π

Solving trigonometric equations Solve the following equations.

sin Θ cos Θ = 0, 0 ≤ Θ < 2π

Solving trigonometric equations Solve the following equations.

sin²Θ = 1/4 , 0 ≤ Θ < 2π

Area of a circular sector Prove that the area of a sector of a circle of radius r associated with a central angle $\(\theta\)$ (measured in radians) is $A=\(\frac\)12r^2\(\theta\)$.

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{Use of Tech} Launching a rocket A small rocket is launched vertically upward from the edge of a cliff $80$ ft above the ground at a speed of $96$ ft/s. Its height (in feet) above the ground is given by $h\(\left\)(t\(\right\))=-16t^2+96t+80$, where $t$ represents time measured in seconds.

a. Assuming the rocket is launched at $t=0$, what is an appropriate domain for $h$?

Draining a tank (Torricelli’s law) A cylindrical tank with a cross-sectional area of $10$ m² is filled to a depth of $25$ m with water. At $t=0$ s, a drain in the bottom of the tank with an area of $1$ m²  is opened, allowing water to flow out of the tank. The depth of water in the tank (in meters) at time $t\(\geq{0}\)$ is $d\(\left\)(t\(\right\))=\(\left\)(5-0.22t\(\right\))^2$.

a. Check that $d\(\left\)(0\(\right\))=25$, as specified.

Draining a tank (Torricelli’s law) A cylindrical tank with a cross-sectional area of $10$ m² is filled to a depth of $25$ m with water. At $t=0$ s, a drain in the bottom of the tank with an area of $1$ m²  is opened, allowing water to flow out of the tank. The depth of water in the tank (in meters) at time $t\(\geq{0}\)$ is $d\(\left\)(t\(\right\))=\(\left\)(5-0.22t\(\right\))^2$.

b. At what time is the tank empty?

Draining a tank (Torricelli’s law) A cylindrical tank with a cross-sectional area of $10$ m² is filled to a depth of $25$ m with water. At $t=0$ s, a drain in the bottom of the tank with an area of $1$ m²  is opened, allowing water to flow out of the tank. The depth of water in the tank (in meters) at time $t\(\geq{0}\)$ is $d\(\left\)(t\(\right\))=\(\left\)(5-0.22t\(\right\))^2$.

c. What is an appropriate domain for $d$?

Yeast growth Consider a colony of yeast cells that has the shape of a cylinder. As the number of yeast cells increases, the cross-sectional area A (in mm²) of the colony increases but the height of the colony remains constant. If the colony starts from a single cell, the number of yeast cells (in millions) is approximated by the linear function N(A) - CₛA, where the constant Cₛ is known as the cell-surface coefficient. Use the given information to determine the cell-surface coefficient for each of the following colonies of yeast cells, and find the number of yeast cells in the colony when the cross-sectional area A reaches 150 mm². (Source: Letters in Applied Microbiology, 594, 59, 2014)

The scientific name of baker’s or brewer’s yeast (used in making bread, wine, and beer) is Saccharomyces cerevisiae. When the cross-sectional area of a colony of this yeast reaches 100 mm², there are 571 million yeast cells.

Demand function Sales records indicate that if Blu-ray players are priced at \$250, then a large store sells an average of 12 units per day. If they are priced at \$200, then the store sells an average of 15 units per day. Find and graph the linear demand function for Blu-ray sales. For what prices is the demand function defined?

Taxicab fees A taxicab ride costs \$3.50 plus \$2.50 per mile. Let m be the distance (in miles) from the airport to a hotel. Find and graph the function c(m) that represents the cost of taking a taxi from the airport to the hotel. Also determine how much it will cost if the hotel is 9 miles from the airport.