asinus

If we are given x such that 25^x - 9^y = 16 and 5^x - 3^y = 2, find x + y.

Hello Guest!

\(25^x - 9^y = 16\\ 5^x - 3^y = 2\)

\(x=1\\ y=1\)

\(x+y=2\)

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On a Cartesian coordinate plane, points (1,2) and (7,-4) are opposite vertices of a square. What is the area of the square?

Hello Guest!

\(A=(y_1-y_2)\times (x_2-x_1)=(2-(-4))\times (7-1) \)

\(A=36\)

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What is the sum of the coordinates of the point you land upon?

Hello Guest!

\(y=\frac{6}{9}x\\ x= \frac{9}{4}=2.25\\ y=\frac{6}{9}\cdot \frac{9}{4}=1.5\\ \color{blue}y+x=1.5+2.25=3.75\)

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What real value of t produces the smallest value of the quadratic t^2 - 9t - 36 + 3t^2 - t + 10?

Hello Guest!

\(f(t)==4t^2-10t-26\\ \frac{df(t)}{dt}=8t-10=0\\ \color{blue}t=\dfrac{10}{8}=1.25\)

The value t = 1.25 produces the smallest value of the quadratic t^2 - 9t - 36 + 3t^2 - t + 10.

The smallest value is - 32.25.

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For what values of y does the following equation have no solutions for x?
xy+6y+20x+4=1

Hello Guest!

\(xy+6y+20x+4=1 \\ x(y+20)=1-4-6y\\ x=-\dfrac{3+6y}{y+20}\)

For  y = - 20 does the equation have no solution for x.

Find the circumradius.

Hello Guest!

Cosine law:

\(cos(\alpha)=\dfrac{b^2+c^2-a^2}{2bc}=\dfrac{2\cdot 25^2-20^2}{2\cdot 25\cdot 25}=0,68\\ \alpha=acos(0,68)=47.156°\\ \color{blue}\frac{\alpha}{2}=23.578°\)

\(cos(\frac{\alpha}{2})=\dfrac{\frac{25}{2}}{r}\\ r=\dfrac{12.5}{cos(23.578°)}=13.639\)

\(The\ circumradius\ is\ \color{blue}r=13.639\).

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What is the greatest possible sum x + y?

Hello Guest!

I only find 8 ordered pairs of integers that satisfy x ^ 2 + y ^ 2 = 6500.

These are:

(10,80) (32,74) (40,70) (56,58) (58,56) (70,40) (74,32) (80,10)

The greatest possible sum x + y is: x + y = 56 + 58 = 114 

or                                                 x + y = 58 + 56 = 114

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The range of the function g(x) = 2/(4 + 2x^2) can be written as an interval (a,b]. What is a + b?

Hello Guest!

\(g(x) = \dfrac{2}{4+2x^2}\)

\(R=b-a\\ b=0.5\\ a\approx 0\ |\ a>0\\ a+b\approx 0.5\ |\ >0.5\)

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(a)  How many chickens are there on Simon's farm?

(b)  Each duck on Thiru's farm lays an average of 6 eggs per week. How many duck eggs are produced on Thiru's farm per week?

Hello KourageKowardlyDog!

\(Simon:\\ (d-84):d=1:7\\ d=7d-7\cdot84\\ 6d=588\)

\(d=98\\ c=14\)

\(Thiru:\\ (d-84):d=2:3\\ 2d=3d-3\cdot 84 \)

\(d=252,\ e=252\cdot 6=1512\\ c=168\)

(a) 14 chickens are there on Simon's farm.

(b) 1512 duck eggs are produced on Thiru's farm per week?

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