assuming the pen is a rectangle, draw a rectangle and draw a vertical line inside the rectangle from top to bottom parallel to the two sides; this divides the pen into two smaller pens
call the horizontal lines the lengths and call the vertical lines the widths
48=2l+3w
divide...

How many 2/3 's are there in 10 ?
10÷(2/3)=10*(3/2)=30/2=15 pieces of pipe

draw a diagram
call the hypotenuse "y"
call the other leg "x"
x+y+11=132
x+y=132-11
x+y=121
x=121-y
use the Pythagorean Theorem
y^2=X^2+11^2
substitute
y^2=(121-y)^2+121
y^2=14,641-242y+y^2+121
cancel...

draw the right triangle
call the vertical leg "a" and call the horizontal leg "b"
one angle is 53º, one angle is 90º, and the third angle is 90-53=37º
sin53=a/45
0.7986=a/45
a=45(0.7986)
a=35.937
cos53=b/45
0.6018=b/45
b=45(0.6018)
b=27...

cscθ/cotθ
(1/sinθ)÷(cosθ/sinθ)
(1/sinθ)*(sinθ/cosθ)
simplify by cancelling the sinθ's
1/cosθ

If v+w=90º (∏/2), then sinv=cosw.
If cosw=5/13, then sinv=5/13.

the largest area is always a square; a square is a special rectangle
200/4=50
you want a square 50 feet by 50 feet
A=502=50*50=2500 square feet
let's try a rectangle 49 by 51: 49*51=2499 square feet

if there are 2 posts, there is 1 group of 85 m
if there are 3 posts, there are 2 groups of 85 m
if there are 4 posts, there are 3 groups of 85 m
if there are 25 posts there are 24 groups of 85 m
24*85=2040 m

2/7 is apple juice
1/2 is orange juice
2/7 + 1/2=4/14+7/14=11/14
14/14 - 11/14=3/14
3/14 is cranberry juice
4/14 is apple juice
7/14 is orange juice
the ratio is 3/14 to 4/14 to 7/14
the ratio simplified is 3:4:7

cscθ=-6/5
cscθ=1/sinθ
sinθ=-5/6
tanθ>0 implies quadrant 3
cos=adjacent/hypotenuse
cosθ=-√(11)/6
cos(θ/2)=-√[(1+cosθ)/2]
cos(θ/2)=-√([1-(√11/6)]/2)
cos(θ/2)=-√([(6-√11)/6]/2)
cos(θ/2)=-√([6-√11]/12)
cos(θ/2)=-(1/2)√([6-√11]/3)

there are 4 90º sectors in a circle
4*64∏=256∏
A=256∏
A=∏r2
r2=256
r=√256
r=16 feet

let x=θ
cotx=1
cotx=1/tanx=cosx/sinx
cosx/sinx=1
sinx and cosx must be equal
this occurs at ∏/4 when sinx=√2/2 and cosx=√2/2
this also occurs at 5∏/4 when sinx=-√2/2 and cosx=-√2/2
cscx=1/sinx
1/sinx=2/√2=2√2/2=√2
1/sinx=-2/√2=-2√2/2=-√2
so cscx=√2 or -√2...

Here's i)
95,658-48,685=46,973
48,685-46,973=1712
now find the GCD of 1712 and 46,973 because 1712 is small enough to factor
1712/2=856
856/2=428
428/2=214
214/2=107
1712=2*2*2*2*107
46,973 is odd and not divisible by 2, so divide 46,973 by 107
46,973/107=439
therefore...

draw a circle with a vertical chord near the left side and a horizontal chord near the bottom of the circle
ab is the vertical chord, from top to bottom, and cd is the horizontal chord from right to left
they intersect in point m
am=6, bm=4, cm=12 and therefore dm=2 because the...

simplify 3√(x+7)-(2/3)(x+7)(3/2)
3√(x+7)-(2/3)(√(x+7)3)
take out an "x+7" in the second term
3√(x+7)-(2/3)(x+7)(√(x+7))
factor out a √(x+7)
(3-[2/3](x+7))(√(x+7))
(3-2x/3-14/3)(√(x+7)
combine 3 and -14/3 (3 becomes...

1 radian=180º/∏ or 180/3.14159=57.295828º
1) 3.2 radians
(3.2)(57.295828)=183.34665º
using only 0.34665º
(0.34665)(60)=20.799'
(0.34665-20/60)(3600)=(0.34665-0.33333)(3600)
(0.01332)(3600)=47.952" or 48"
183º 20' 48"
4) 6 radians
6(180/3...

a(n)=(a)(r)(n-1) where a=1st term and r=common ratio
a(1)=(7)(2)(1-1)=7*20=7*1=7
a(2)=(7)(2)(2-1)=7*2=14
a(3)=(7)(2)(3-1)=7*22=7*4=28
a(4)=(7)(2)(4-1)=7*23=7*8=56
and so on

A=L*W
A=21 ft2
21=7*3
P=2L+2W
20=2L+2W
10=L+W
10=7+3
the dimensions are 3 ft by 7 ft

draw a diagram
point A is (-5,-12)
assuming that the distances to the axes are perpendicular to the axes...
the midpoint of the segment from point A to the x-axis is (-5,-6)
the midpoint of the segment from point A to the y-axis is (-2.5,-12)
connect these points to form a right...