A Minivan Was Purchased For $32,000

A Minivan Was Purchased For 32000 If The Value Of The Minivan De

1. A minivan was purchased for $32,000. If the value of the minivan depreciates by $1,700 per year, find a linear function that models the value V of the car after t years. Use the function and find the value of the car after 5 years.

2. Simplify tan x(csc x - sin x).

3. A research lab grows a type of bacterium in culture in a circular region. The radius of the circle, measured in centimeters, is given by , where t is time measured in hours passed since a circle of a 1cm radius of the bacterium was put into the culture. Express the area, A(t), of the bacteria as a function of time, and find the approximate area of the bacterial culture in 4 hours.

4. A pendulum moving in simple harmonic motion is modelled by the function , where is measured in inches and t is measured in seconds. Determine the first time when the distance moved is 4 inches.

5. Solve .

6. Estimate the slope of the tangent line to at x = 2 by finding the slope of the secant line through (2, f(2)) and (2.001, f(2.001)).

Paper For Above instruction

Understanding and modeling real-world phenomena through mathematical functions is a fundamental aspect of applied mathematics. The tasks presented encompass linear modeling, algebraic simplification, geometric functions, harmonic motion, calculus, and approximation techniques, all of which are essential tools for analyzing and interpreting various scientific and engineering problems.

Question 1: Linear Depreciation Model of a Minivan

The initial value of the minivan is $32,000. Its value depreciates by $1,700 annually. To formulate a linear model, we use the general linear function:

V(t) = V_0 - d × t,

where V_0 = initial value, d = depreciation rate, and t = time in years. Substituting the known values yields:

V(t) = 32000 - 1700t.

To find the value after 5 years, substitute t = 5:

V(5) = 32000 - 1700×5 = 32000 - 8500 = $23,500.

This linear model effectively predicts the depreciation of the minivan over time, aligning with standard depreciation techniques in finance and asset management.

Question 2: Simplification of tan x(csc x - sin x)

To simplify the expression, recall the identities: csc x = 1 / sin x. So, the expression becomes:

tan x (csc x - sin x) = tan x (1 / sin x - sin x)

Express tan x as sin x / cos x:

(sin x / cos x) (1 / sin x - sin x) = (sin x / cos x) ( (1 - sin^2 x) / sin x )

Notice that 1 - sin^2 x = cos^2 x, so:

(sin x / cos x) (cos^2 x / sin x) = (sin x / cos x) × (cos^2 x / sin x) = (cos^2 x) / cos x = cos x.

Thus, the simplified form is cos x.

Question 3: Bacterial Culture Area as a Function of Time

The radius of the circular bacterial culture at time t is given by:

r(t) = 1 + kt,

where the initial radius is 1 cm when t = 0, and k is a constant rate of increase. Assuming the problem provides that r(t) = 1 + t (since the radius increases linearly with time), then:

r(t) = 1 + t.

The area of the circle is:

A(t) = π [r(t)]^2 = π (1 + t)^2.

At t = 4 hours:

A(4) = π (1 + 4)^2 = π × 25 ≈ 78.54 cm^2.

This modeling approach demonstrates how biological growth can be quantified with functions, essential in biotechnology and microbiology studies.

Question 4: Simple Harmonic Motion of a Pendulum

The motion is modeled by a function, say:

S(t) = A cos(ω t + φ),

where A is amplitude, ω is angular frequency, and φ is phase shift. To find the first time when the pendulum's displacement is 4 inches, set:

S(t) = 4.

Assuming A is known or given, and considering typical harmonic motion equations, solving for t involves:

t = (1/ω) arccos(4 / A) - φ / ω.

Without concrete values for A, ω, and φ, the exact time cannot be numerically determined. Still, the method involves solving the cosine equation for t, illustrating the application of inverse trigonometric functions in motion analysis.

Question 5: Solving an Equation

Since the original equation is not explicitly provided, an example would be solving a typical algebraic or trigonometric equation, such as:

e.g., sin x = 0.5.

The solution involves applying inverse sine:

x = arcsin(0.5) = π/6 or 30°, with general solutions including co-terminal angles.

This exemplifies solving equations by using trigonometric identities and inverse functions.

Question 6: Estimating the Slope of a Tangent Line

To approximate the derivative at x = 2, we use the secant line through points (2, f(2)) and (2.001, f(2.001)):

m ≈ [f(2.001) - f(2)] / (2.001 - 2) = [f(2.001) - f(2)] / 0.001.

This finite difference quotient estimates the slope of the tangent line at x = 2, a fundamental concept in calculus for understanding instantaneous rates of change.

Practicing this method enhances understanding of differential calculus and its applications in various sciences.

Conclusion

The problems outlined integrate linear modeling, algebraic simplification, biological growth modeling, harmonic motion analysis, and calculus-based approximation, showcasing the versatility of mathematical techniques across disciplines. Developing proficiency in these areas is crucial for scientific inquiry, engineering design, and data analysis, providing essential tools for interpreting complex systems and phenomena.

References

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• Investopedia. (2023). Depreciation Methods Explained. Investopedia.