Answer:
The number of beads used must be at least 40 to make a decorated rope of at least 20 feet length. The inequality is:
Step-by-step explanation:
Let the number of beads used be 'n'.
Now, length of 1 bead = 6 in
∴ Length of 'n' beads = 
Now, minimum length of rope is 20 ft. Converting feet to inches, we get:
Now, beads are inserted in the rope. So, the length of rope is same as the length of all the beads joined together.
As per question, length must be at least 20 ft or 240 in.
Therefore, the length of all beads joined together must be at least 240 inches. This gives,
Therefore, the number of beads used must be at least 40 to make a decorated rope of at least 20 feet length.
He had 44 cookies because if he gave out half of them then you would add what he had left after he gave the other half of them away which 14+ 8 is 22 and 22×2 is 44 so the answer is 44 cookies
Let's work on the left side first. And remember that
the<u> tangent</u> is the same as <u>sin/cos</u>.
sin(a) cos(a) tan(a)
Substitute for the tangent:
[ sin(a) cos(a) ] [ sin(a)/cos(a) ]
Cancel the cos(a) from the top and bottom, and you're left with
[ sin(a) ] . . . . . [ sin(a) ] which is [ <u>sin²(a)</u> ] That's the <u>left side</u>.
Now, work on the right side:
[ 1 - cos(a) ] [ 1 + cos(a) ]
Multiply that all out, using FOIL:
[ 1 + cos(a) - cos(a) - cos²(a) ]
= [ <u>1 - cos²(a)</u> ] That's the <u>right side</u>.
Do you remember that for any angle, sin²(b) + cos²(b) = 1 ?
Subtract cos²(b) from each side, and you have sin²(b) = 1 - cos²(b) for any angle.
So, on the <u>right side</u>, you could write [ <u>sin²(a)</u> ] .
Now look back about 9 lines, and compare that to the result we got for the <u>left side</u> .
They look quite similar. In fact, they're identical. And so the identity is proven.
Whew !
