EnchantedLava68

Because Dylan descends 60 feet in 20 minutes, he will descend 3 feet 1 minute.  From this, we have Dylan's height = 400-3m, where m is the number of minutes after he starts his descent.  

With this equation, he will reach the ground when 400-3m=0, or when 3m=400, simplifying to after 400/3 minutes, or 133 minutes 20 seconds.  

So for this question, formulas for volume and surface areas of spheres are essential.  The volume of P is defined as \(\frac{4}{3}πr^3 = \frac{4}{3}π36^3\), and Surface area of P is \(4πr^2\), which in this case is equivalent to \(4π36^2\).  Once we put this into the inequality (volume of P)/(surface area of P) ≤ t, we have, after simplification 12≤t.  This, the smallest real number t should be 12.  

Since the graph passes through (-3, -2), (-1, 0), and (-5, 0), we have the equations \(9a-3b+c = -2, a-b+c = 0,\) and \(25a-5b+c= 0\).  Solving, we have a = 1/2, b = 3, c = 5/2, which means that abc = 15/4

a = 6

b = -5

c = 4

Let the age of the first elephant be a

Let the age of the second elephant be b

Let the age of the tortoise be x.  

From the given information, we have 

x = a + b + 52

x+10 = 2(a+10+b+10)

Simplifying both, we have x=a+b+52 and x+10 = 2(a+b)+40, x = 2(a+b) + 30.  

Subtracting the two, we have a+b-22 = 0, so a+b = 22.  

This means that x = a+b +52, or 22+52, or 74 years old.  

Let the dimensions of the package be \(a \) by \(b\) by \(c \).  Thus, the volume is \(abc\). 

The total material required to create the package would be the surface area, namely, \(2(ab+bc+ac)\).  With this, we seek to find an inequality that links \(ab+bc+ac\) to \(abc\), and the AM-GM Inequality does just that.  

By the AM-GM Inequality for 3 variables, \(\frac{ab+bc+ac}{3}≥\sqrt[3]{abc^2}\), so \(\frac{ab+bc+ac}{3}≥\sqrt[3]{1440000}\).  

Simplifying, we have \(ab+bc+ac≥60\sqrt[3]{180}\).  The minimum namely, the right side of the above equation is achieved when the equality condition of the AM-GM is met, or when ab=bc=ac.  This simplifies to a=b=c, or a=b=c= \(2\sqrt[3]{150}\), from abc = 1200

My answer for the dimensions would be \(2\sqrt[3]{150}\) by \(2\sqrt[3]{150}\) by \(2\sqrt[3]{150}\).

\(\frac{x+2}{4x+8}-\frac{3-x}{x^2+2x}\)

= \(\frac{(x+2)(x^2+2x)}{(4x+8)(x^2+2x)}-\frac{(3-x)(4x+8)}{(x^2+2x)(4x+8)}\)

= \(\frac{x^3+4x^2+4x}{(4x+8)(x^2+2x)}-\frac{-4x^2+4x+24}{(x^2+2x)(4x+8)}\)

= \(\frac{x^3+8x^2-24}{(x^2+2x)(4x+8)}\)

= \(\frac{(x+2)(x^2+6x-12)}{x(x+2)(4x+8)}\)

= \(\frac{x^2+6x-12}{4x^2+8x}\)

Restrictions: X cannot equal 0, -2, as these values would make the denominator 0.  

Because \(f^{-1}(f(x)) = x\), \(f^{-1}(f(3)) = 4\), simplifying to \(f^{-1}(4) = 3\).  

Thus, \(f(5) + f^{-1}(4) = 1+3 = 4\).  

The answer is 4.

Because f(x) is the inverse of \( f^{-1}(x)\), and g(x) = \( f^{-1}(x)\), f(x) is the inverse of g(x).  Thus, if g(15) = 0, g(0) = 4, g(3) = 9, g(9) = 3, then f(0) = 15, f(4) = 0, f(9) = 3, f(3) = 9.  

Thus, f(f(9)) = f(3) = 9

From 1-50, the numbers that are congruent to 2 (mod 8) are 2, 10, 18, 26, 34, 42, and 50.  Altogether, this totals as 7 numbers.  With there being 50 numbers between 1 and 50, the probability that the tile is blue (congruent to 2 mod 8) is 7/50, or 14%.