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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...

There are six outcomes for the cube and five outcomes for the spinner, for a total of 6*5=30 outcomes (1-1, 1-2,..., 6-5).   In six of those outcomes the spinner spins a 5 (1-5, 2-5,...,6-5), and in five outcomes the cube rolls a 5 (5-1, 5-2, ..., 5-5). There is exactly one outcome...

(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...

There are many functions that alternately increase and decrease (called periodic functions), but the simplest of these is the sine function, y = A sin( (2pi/T) t - φ ) + b.   Here y and t are the variables: magnitude and time in your case. A is called the amplitude; it...

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

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...

(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...

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...

(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°.

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...

Position: r = [t2+1, ln(2t+3)]   Velocity: v = dr/dt = [2t, 2/(2t+3)]   Acceleration: a = dv/dt = [2, -4/(2t+3)²]

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) +...

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...

Pattern 1: 1+2 = 3 dots Pattern 2: 1+2+3 = 6 dots Pattern 3: 1+2+3+4 = 10 dots ... Pattern n: 1+2+3+4+...n+(n+1) = ∑k=1n+1 (k) = (n+1)(n+2)/2 dots   To find n for 1953 dots, set (n+1)(n+2)/2...

This pattern formation is an iterative (recursive) process and leads to a so-called fractal. In step 0 you start with one 3-segment triangle. Then in each following step you take four of the triangles of the previous step and put them together to form a new triangle (with one of the four triangles...

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...

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 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...

1. Find the x- and y-components of both forces. 2. Add the x- and y-components separately. 3. Find the total force (magnitude and direction).   1.    F1x = 230 cos(125)     F1y = 230 sin(125)        F2x = 175 cos(60)...

You find the launching angle θ from the relation tan (θ) = v0y/v0x, where v0y and v0x are the y- and x-components of the initial velocity whose magnitude is v0= sqrt(v0y²+v0x²). We need to use the kinematic equation that does not contain time, v² = v0² - 2gy = v0y²+v0x²...