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## Philip P.'s Resources

Your equations are correct:   f(x) = (\$4.40)x + \$50 g(x) = (\$3.90)x + \$90   x represents the number of mugs you purchase and f(x) and g(x) represent your cost.   (c) x = 120 mugs        f(120) = (\$4.40)(120) +...

The money is divided into 1 + 3 + 1/2 = 4 1/2 parts.  The smallest part is the 1/2 part, which equals \$42.50.   (1/2)/(41/2) = \$42.50/x   1/9 = \$42.50/x   x = (\$42.50)*9 = \$382.50

The results of a survey showed that 3 out of 10 people had voted for Lucky Luke. If there were 40,000 votes: a) what is the ratio of those who voted for Lucky Luke to those who did not?  (3/10)/(7/10) = 3/7 b) How many voted for Luke?  3/10 = x/40,000.  Solve for...

Let x = the numerator.  The denominator exceeds the numerator by 7, so it's x+7.  The full fraction is:        x/(x+7)   If 3 is added to the numerator and 1 is subtracted from the denominator, it's equal to 4/5      ...

∫(2√x)/(√x) dx   Note that √x = x1/2.  Now let u = x1/2: du = (1/2)x-1/2 dx = (1/2u) dx. Hence 2u du = dx.    Now substitute:         ∫(2u/u) 2u du = 2 ∫ 2u du   Note that 2u = eln(2^u) = euln(2)        ...

I assume the expression is 1/(x2+y).  In math division by zero is undefined.  So when you have a rational expression like this, the expression will be undefined when the denominator is zero.  The denominator is (x2+y), so we need to know the values of x and y that can make...

Let x = the unknown number(s).     (x + 6)2 = 169   x2 + 12x + 36 = 169   x2 + 12x - 133 = 0   (x+19)(x-7) = 0   x = -19 and 7

Use the "vertex form" of the quadratic equation:        y = a(x-h)2 + k   where (h,k) is the location of the vertex.  In this problem (h,k) is given as (3,-4) so:        y = a(x-3)2 - 4   To...

2x + x + 4 = -17   Subtract 4 from both sides:        2x + x + 4 - 4 = -17 - 4   Combine like terms:        3x = -21   Divide both sides by 3:       ...

(3x5y3)3 (5xy7)2        = 33(x5)3(y3)3(5)2*(x)2(y7)2        = 27x15y925x2y14        = (27*25)(x15+2)(y9+14)        = 675x17y23

y= x3 - 11x2 + 24x + 5   dy/dx = d(x3)/dx - d(11x2)/dx + d(24x)/dx + d(5)/dx   dy/dx = 3x2 - 22x + 24

The Coulomb Force is an inverse square force; that is, it's proportional to 1/r2 where r is the distance between the particles.   F = (1/4piε0)*q1q2/r2   so when you cut the distance in half, the force increases by a factor of four: 1/(1/2)2 = 1/(1/4) = 4.

A direct variation equation has the general form:        y = kx   where k is the constant of variation.  Since k is given as -8, the direct variation equation in this problem is:        y = -8x   For...

Let x = the unknown number The number divided by 4:  x/4 equals 6: = 6   x/4 = 6   Multiply both sides by 4:   x = 6*4   x = 24

log3(9/√3)        = log3(32/31/2)   Now use the log rule log(a/b) = log(a) - log(b)        = log3(32) - log3(31/2)        = 2 - 1/2 = 3/2

Assuming that 17,561,600 is the number of workers in 2008, then the equation would be:        W(t) = (322,560)*t + 17,561,600   Where t is the number of years since 2008.  In 2018, t = 2018-2008 = 10.  Plug t= 10 into the above equation...

P(t) = P0*ekt   Where: P(t) = the population at time t P0 = the initial population k = the exponential growth rate t = time e = Euler's Number ≅ 2.718   If the population doubles when t = 62 years, then P(t=62) = 2P0.        ...

2x - 2y = -3 -4x + 4y = 6   Multiply the first equation by 2:   2*(2x - 2y = -3) = 4x - 4y = -6   Now add this to the second equation:     4x -  4y = -6  -4x + 4y =  6 -----------------   ...

Percent Increase = (100%)* (Final Amount)/(Starting Amount)                              = (100%)*(1,900,000)/(200,000)                             ...

f(x) = (4x-6)/√(8-x)   In math, division by zero is undefined.  So any value of x that makes the denominator of f(x) equal zero will make f(x) undefined.  All such values of x are excluded from the domain of f(x).  The denominator of f(x) is √(8-x), and √(8-x) = 0 when...