(a) The equation in slope-intercept form for the money raised biking M miles, that factors in all costs is:
y = 1.50M - 240
where y is the total money raised (including the $240 in costs) and M is the number of miles biked.
(b) Mariela and Jelena will bike a total of 680 miles (340 miles each way) to LA and back. Thus, the total money raised will be:
y = 1.50(680) - 240 = $840
Therefore, Mariela and Jelena will raise $840 for charity once they get to LA.
(c) Given that sin(θ) and is in Quadrant II, we know that sin(θ) is positive and cos(θ) is negative.
Using the half-angle formulas, we can find:
sin(θ/2) = ±sqrt((1 - cos(θ)) / 2) = ±sqrt((1 - sqrt(1 - sin²(θ))) / 2)
cos(θ/2) = ±sqrt((1 + cos(θ)) / 2) = ±sqrt((1 + sqrt(1 - sin²(θ))) / 2)
tan(θ/2) = sin(θ) / (1 + cos(θ)) = sqrt(1 - sin²(θ)) / (1 - sqrt(1 - sin²(θ)))
Since sin(θ) is positive and cos(θ) is negative in Quadrant II, we know that:
sin²(θ) + cos²(θ) = 1
sin²(θ) + (-cos(θ))² = 1
sin²(θ) + cos²(θ) = 1
Thus, we can find:
sin(θ/2) = sqrt((1 - sqrt(1 - sin²(θ))) / 2) = sqrt((1 - sqrt(1 - (-0.8)²)) / 2) = sqrt((1 - sqrt(0.36)) / 2) = sqrt(0.32) ≈ ±0.5657
cos(θ/2) = -sqrt((1 + sqrt(1 - sin²(θ))) / 2) = -sqrt((1 + sqrt(1 - (-0.8)²)) / 2) = -sqrt((1 + sqrt(0.36)) / 2) = -sqrt(1.18