geno3141

There are 5280 feet in each mile.

There are 60 seconds in each minute and 60 minutes in each hour so there are 3600 seconds in each hour.

(60 miles / hour) x (5280 feet / 1 mile) x (1 hour / 3600 seconds) 

=  (60 x 5280 feet) / (3600 seconds)  =  88 feet / second 

The point where the two graphs cross is the solution.

If the graphs cross at the point (6, -5), then the solution is:  x = 6 and y = -5.

If the lines are parallel, they don't cross and there is no solution.

If, when graphed, one line sits exactly on top of the other, then there is no single solution; every point is a solution to both equations.

ln(x) - ln(10)  =  ln(9)

Two log expressions that create a subtraction problem can be replaced with one log expression of a division.

--->  ln(x) - ln(10)  =  ln(x/10)

So:  ln(x/10)  =  ln(9)

This statement is true only if  x/10  =  9   --->  x = 90

If you know that the triangle is a right triangle, use the Pythagorean Theorem.

If you don't know that it is a right triangle, you can't find the third side unless you have more information; for instance, the size of the angle between the two sides that you know.

√(-12)  =  √(4 x 3 x -1)  =  √(4) x √(3) x √(-1)

√(4)  =  2

√(-1)  can be replaced with the symbol  i:

   =  2√(3)i  or  2i√(3) 

sin²(x) + cos²(x)  =  1

sec²(x)  =  1 + tan²(x)

csc²(x)  =  1 + cot²(x)

[sin²(x) + cos²(x)] + tan²(x) + cot²(x) + [sec²(x)] + [csc²(x)]  =  7

--->   1 + tan²(x) + cot²(x) + [1 + tan²(x)] + [1 + cot²(x)]  =  7

--->   3 + 2tan²(x) + 2cot²(x)  =  7

--->   2tan²(x) + 2cot²(x)  =  4

--->   tan²(x) + cot²(x)  =  2

--->   tan²(x) + 1 / tan²(x)  =  2

--->   tan⁴(x) + 1  =  2tan²(x)

--->   tan⁴(x) - 2tan²(x) + 1  =  0

--->   (tan²(x) - 1)(tan²(x) - 1)  =  0

--->   tan²(x)  =  1

--->   tan(x)  =  1    or    tan(x)  =  -1

--->   x  =  π/4  or  5π/4     or     x  =  3π/4  or  7π/4

log₃(6x + 5)  =  2

Write into exponential notation:  3²  =  6x + 5

--->   9  =  6x + 5

--->   4  =  6x

--->   x  =  4/6  =  2/3

Since  y  =  (1/4)x + 2  and  y  =  x - 1   --->   (1/4)x + 2  =  x - 1

Multiply all the terms by 4   --->   x + 8  =  4x - 4

Subtract x from both sides  --->         8  =  3x - 4

Add 4 to both sides            --->        12  =  3x

Divide both sides by 3        --->          x  =  4

A right triangle will be formed if a² + b²  =  c².

The longest side must become the value of c; so is:  30² + 122²  =  125² ?

The angle at the rear end of the dump truck must be at least 45°.

For that angle, a right triangle has been formed where the front end is the opposite side and the length of the load is the hypotenuse.

When the front end of the dump truck has been raised 8', the opposite side is 8 while the hypotenuse is 14.

opposite divided by hypotenuse is sine:  sin = opp / hyp  =  8/14.

The angle = sin^-1(8/14) = 34.8°.

If you haven't had any trig, this can also be solved by similar triangles. Post back if you need that solution.