Solution Procedure
Considering the spool as a cylindrical shape, and calculating the circumference based on the area cited, we establish the connection between circumference and area as follows:
... C = 2√(πA)
To find ten times this circumference, we calculate:
... 10C = 20√(π·20 cm²) = 40√(5π) cm ≈ 159 cm
Derivation of Formulas
The standard formulas for circumference and area are:
... C = 2πr
... A = πr²
Multiplying the area formula by π and extracting the square root yields:
... πA = (πr)²
... √(πA) = πr
Doubling this value results in the circumference formula:
... C = 2√(πA) = 2πr
Assuming arcs are measured in degrees, let S represent the following sum:
S = sin 1° + sin 2° + sin 3° +... + sin 359° + sin 360°
By rearranging, S can be reformulated as
S = [sin 1° + sin 359°] + [sin 2° + sin 358°] +... + [sin 179° + sin 181°] + sin 180° +
+ sin 360°
S = [sin 1° + sin(360° – 1°)] + [sin 2° + sin(360° – 2°)] +... + [sin 179° + sin(360° – 179)°]
+ sin 180° + sin 360° (i)
However, for any real k,
sin(360° – k) = – sin k
Thus,
S = [sin 1° – sin 1°] + [sin 2° – sin 2°] +... + [sin 179° – sin 179°] + sin 180° + sin 360°
S results in 0 + 0 +... + 0 + 0 + 0 (... since sine of 180° and 360° are both equal to 0)
Therefore, S equals 0.
Each pair within the brackets negates itself, leading the sum to total zero.
∴ sin 1° + sin 2° + sin 3° +... + sin 359° + sin 360° equals 0. ✔
I hope this clarifies things. =)
Tags: sum summatory trigonometric trig function sine sin trigonometry
Answer with explanation: Given that Circle 1 has a center at (−4, −7) and a radius of 12 cm, while Circle 2's center is at (3, 4) with a radius of 15 cm. Two circles are similar if one can be transformed and scaled to fit over the other, creating identical circles. The circles qualify as similar because the transformation rule (x,y) → (x+7,y+11) can be applied to Circle 1, followed by dilation using a scale factor of 5/4. Since Circle 1's center is at (-4,-7), we translate it to (3,4) through (-4+7,-7+11). With Circle 1 having a radius of 12 and Circle 2 having 15 cm, we denote the scale factor as k.
Let’s determine the actual mean
First, we sum all the values
87+46+90+78+89 = 390
Then, we divide 390 by the count of numbers present.
390/5 = 78
Thus, the mean is 78
Emi did not manage to calculate the difference
