Explanation

The frequency heard by the observer as the car approaches is given by the Doppler effect formula:

f' = f * (v / (v - v_s))

where f = 500 Hz, v = 340 m/s, and v_s = 20 m/s.

f' = 500 * (340 / (340 - 20))

= 500 * (340 / 320)

= 500 * 1.0625

= 531.25 Hz

Explanation

The relativistic length contraction formula is:

L = L₀√(1 - v²/c²)

Given L = L₀/2:

L₀/2 = L₀√(1 - v²/c²)

1/2 = √(1 - v²/c²)

1/4 = 1 - v²/c²

v²/c² = 3/4

v = √(3/4) * c = √3/2 * c

The answer is v = √3/2c.

Explanation

The sound level β (in dB) is given by:

β = 10 log(I / I₀)

where I₀ is the threshold intensity (faintest audible sound). Given I = I₀:

β = 10 log(1) = 0 dB

Explanation

Let's denote the mass and radius of both objects as M and R, respectively.

For a thin-walled cylinder about its central axis, I_cylinder = MR².

For a solid sphere about its diameter, I_sphere = (2/5)MR².

The ratio is I_cylinder / I_sphere = MR² / ((2/5)MR²) = 5/2.

Explanation

After 3 half-lives, the mass remaining is calculated as:

Mass remaining = Initial mass / 2³

= 160 μg / 8

= 20 μg

Explanation

In the quadratic equation ax² + bx + c = 0, a cannot be zero.

If a = 0, the equation becomes linear (bx + c = 0), and it's no longer a quadratic equation.

Explanation

For matrices A and B, the inverse of the product AB is given by:

(AB)⁻¹ = B⁻¹A⁻¹

Explanation

The derivative of y = x² is dy/dx = 2x.

At x = 2, the slope is 2(2) = 4.

The tangent line equation is y - 4 = 4(x - 2), which simplifies to y = 4x - 4, or 4x - y = 4.

Explanation

To determine the type of conic section, rewrite the equation in standard form:

9x² + 4y² = 36

Divide by 36:

x²/4 + y²/9 = 1

Since 9 > 4, the major axis is vertical.

Explanation

A critical point occurs where the derivative of a function is zero or does not exist.

These points are important for finding local maxima, minima, or points of inflection.