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Answers by Andre W.

can you teach me tenses? all of them? please (answer)

In English, there are 6 past tenses, 6 non-past tenses, and 4 conditional tenses.   Past tenses. 1. Simple past: I ate. 2. Past continuous: I was eating. 3. Simple present perfect: I have eaten. 4. Perfect continuous: I have been eating. 5. Simple past perfect: I...

Chemistry problem about solutions. (answer)

(a) A 0.3 M solution of perchloric acid contains 0.3 mol/L, or .15 mol for .5 L. The molar weight of perchloric acid (HClO4) is 100.46 g/mol, so we need 100.46 g/mol (.15 mol) = 15.069 g of perchloric acid. With a density of 1.47 g/mL, this corresponds to a pure volume of 15.069 g/(1.47...

mathematical induction (answer)

Let's first write down the first few decompositions to see the pattern: 14 = 2(3)+1(8)            21 = 7(3) 15 = 5(3)                    ...

Anti- Derivatives. (answer)

Average velocity: (1/5) ∫05 (6-4t) dt = (1/5) [6t-2t²]05 = -4 m/s   Average speed: (1/5) ∫05 |6-4t| dt = (1/5) ( ∫01.5 (6-4t) dt - ∫1.55 (6-4t) dt ) = (1/5) ( [6t-2t²]01.5 - [6t-2t²]1.55 ) = 5.8 m/s   Note: average speed is based on distance traveled, which is always...

Solve the equation (logarithms) (answer)

(i) log2(x3+1) - 2 log2x = log2(x2-x+1) -2 log2 ((x³+1)/(x²(x²-x+1))) = -2 Now use long division to factor (x³+1) as (x+1)(x²-x+1): log2 ((x+1)/x²) = -2 (x+1)/x² = 2-2 x+1 = (1/4)x² x² - 4x - 4 = 0   Solve this quadratic...

Trigonometry (answer)

8 csc(2x) cot(2x) = 3 8 cos(2x)/sin²(2x) = 3 8 cos(2x)/(1 - cos²(2x)) = 3   Let u = cos(2x):   8 u/(1-u²) =3 8u = 3(1-u²) 3u² + 8u - 3 = 0   Solve this quadratic equation, get u = 1/3, -3. Since abs(cos(2x))≤1, u = 1/3 is the only valid...

Trigonometry (answer)

(i) 2 sin x cos x - cos x + 4 sin x -2 = (cos x + 2)(2 sin x - 1)   (ii) (cos x + 2)(2 sin x - 1) = 0 ⇒ sin x = 1/2 ⇒ x = 30°, 150°, -330°, -210°.

Logarithms (answer)

Use the fact that logba = 1/logab. Let x = logab = 1/logba, so your equation becomes 2 x + 4/x = 9. Multiply by x: 2x² -9x + 4 = 0. Solve this quadratic equation, get x = 1/2, 4. Since a>b, x = logab <1, so only x=1/2 is a valid solution. Now solve logab...

Find the limit? (answer)

Use the trig identity sin(x+h) = sin(x)cos(h) + cos(x)sin(h), so that   (sin(x+h)-sin(x))/h = (sin(x)(cos(h) - 1) + cos(x)sin(h))/h.   Now take the limit   limh→0 (sin(x)(cos(h) - 1) + cos(x)sin(h))/h = sin(x) limh→0 ((cos(h) - 1)/h) +...

Integration by parts of Erf(x) (answer)

The integration-by-parts formula is ∫u v' dx = uv - ∫u' v dx   so with u =erf(x), u' = 2e-x²/√π, v' = 1, and v = x, you get ∫erf(x) dx =  x erf(x) - (2/√π) ∫xe-x² dx = x erf(x) + (1/√π) e-x² , where I used a u-substitution (u=e-x²) to evaluate the right-hand-side...

Can you reconcile this energy (answer)

The total energy of the system basketball+rubber ball is conserved in this process, so the total initial potential energy equals the total final potential energy (the initial and final kinetic energies are zero). However, right after the two balls bounce, the larger basketball will collide...

whats the momentum (answer)

Momentum is mass times velocity. The total momentum of the system bug-car stays the same during the collision, so the change in the car's momentum is equal and opposite to the change in the bug's momentum.   Since the car's mass is much higher than the bug's mass, the bug's velocity...

Find its volume? (answer)

Find the infinitesimal volume of one cross-sectional slice of thickness dx, width y=e-x, and, since it is a square, height also e-x : dV = (e-x)(e-x) dx = e-2x dx   Integrate from 0 to 3 to find the total volume: V =∫03 e-2x dx = [-(1/2) e-2x]03 = -(1/2) (e-6 - 1) = (1 - e-6)/2...