Containers for liquids have interior volumes that are larger than their rated capacity. This is necessary to allow for expansion of the liquid contents as the temperature changes. You have a five gallon steel fuel can manufactured to have an interior volume of 5.22 gallons (19.76 L). The coefficient of thermal expansion of steel is small enough that this volume will not change significantly with temperature in this problem.

One cold spring morning, you pump 5.00 gallons (18.93 L) of gasoline into your fuel can, leaving 0.22 gallons (0.83 L) unfilled. The gasoline comes from an underground tank at 54.9°F (12.7°C, 285.9 K), but the outside temperature is 31°F (-0.6°C, 272.6 K). You then screw on the cap tightly and leave the can outside. The weather forecast calls for the temperature to stay constant in the morning morning, followed by a significant warming trend in the afternoon, when the temperature will rise to a high of 75°F (23.9°C, 297.0 K). The atmospheric pressure will stay constant at 973 mbar (97.3 kPa) all day.

1. Gasoline has a coefficient of thermal expansion of 9.50 x 10^-4/°C. What will the unfilled volume in your fuel can be after the gasoline cools in the morning?

2. What will the absolute pressure of the air in the sealed fuel can be then?

3. What will the unfilled volume in your fuel can be when the gasoline and air in the fuel can warm to the afternoon high temperature?

4. What will the absolute pressure of the air in the sealed fuel can be then, assuming no air can escape?

5. Your fuel can actually has a safety gasket which will allow air to leave the can if the pressure difference to the outside (called the “gauge pressure”) exceeds 5.0 psi (34 kPa). Will the safety gasket vent air from the fuel can in the afternoon?

The concept of “market equilibrium” is defined as when the quantity of a commodity demanded is equal to the quantity supplied. Assume that the demand function and the supply function are both linear. How can good advertising affect market equilibrium? How can bad advertising affect market equilibrium?

solve y" + 4y = 5 tan(2x)

Use power series to find two linearly independent solutions centered at the point x=0

1) y'' + 2y' - 2y = 0

2) 2x²y'' + x(x-1)y' - 2y = 0

please show work, thank you!

Solve the given Boundary Value Problem. Apply the method undetermined coefficients when you solve for the particular solution.

y′′+2y′+y=(e^-x)(cosx−sinx)

y(0)=0,y(π)=e^π

Define a function ?∶ ℝ→ℝ by

?(?)={?+1,[?] ?? ??? ?−1,[?]?? ????

where [x] is the integer part function. Is ? injective?

1. (b) Verify if the following function is bijective. If it is bijective, determine its inverse.

?∶ ℝ\{5/4}→ℝ\{9/4} , ?(?)=(9∙?)/(4∙?−5)

1. Let q and p be natural numbers, and show that the metric on R^q+p is equivalent to the metric it has if we identify R^q+p with R^q × R^p.

Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) 2 0 e^x/ 1 + x^2 dx, n = 10 (a) the Trapezoidal Rule (b) the Midpoint Rule (c) Simpson's Rule

Solve the following differential equation: y''+4y'+4y=u(t-1)-u(t-3), y(0)=0, y'(0)=0

Solve the wave equation
∂2u/∂t2 = 4 ∂2u/∂x2 , 0 < x < 2, t > 0

subject to the following boundary and initial conditions.
u(0, t) = 0, u(2, t) = 0,
u(x, 0) = { x, 0 < x ≤ 1
2 − x, 1 < x < 2 , ut(x, 0) = 0

Define our “universe” S as the region in the xy-plane that is described by S = {(x, y) : |x| + |y| ≤ 2 and y > −1} .

(a) Mathematically and graphically describe three sets A1, A2, and A3 taken from S that are mutually exclusive but not collectively exhaustive.

(b) Mathematically and graphically describe three sets B1, B2, and B3 taken from S that are collectively exhaustive but not mutually exclusive.

(c) Mathematically and graphically describe three sets C1, C2, and C3 that form an event space for S. Make the area of C2 equal twice the area of C1, and make the area of C3 equal to three times the area of C1.

Draw a quick but accurate sketch of f(x) = √x²−4 over the interval [−4,0]. This covers the interval of integration.

1. Partition the interval of integration into 10 intervals. Show this on your graph with a right or left Riemann Sum

2. Create a table showing your interval index, i, the value x_i at which you evaluate f(x) in each interval, the values of f(x_i) and ∆x for each interval, and the contribution each rectangle makes toward the Riemann Sum. Evaluate the Riemann Sum for f(x) over the integration interval using your partition.

Using the method of separation of variables and Fourier series, solve the following heat
conduction problem in a rod.
∂u/∂t =∂2u/∂x2
, u(0, t) = 0, u(π, t) = 3π, u(x, 0) = 0

Using variation of parameters, find a particular solution of the given differential equations:

a.) 2y" + 3y' - 2y = 25e^-2t (answer should be: y(t) = 2e^-2t (2e^{5/2 t} - 5t - 2)

b.) y" - 2y' + 2y = 6 (answer should be: y = 3 + (-3cos(t) + 3sin(t))e^t )

Please show work!