Final answer:
The equation for the initial path of the rock released at a –45-degree angle from a slingshot, which forms a tangent to a circle with radius 1.25 feet, is in standard form Ax + By = C, where A = 1, B = 1, and C = 0.
Thus, the equation is x + y = 0.
Step-by-step explanation:
To write the equation for the initial path of the rock released from a slingshot, we must understand that the path will be a tangent to the circle described as the slingshot is spun. Given the scenario where the rock is released at a –45-degree angle to the horizontal, the tangent line at the point of release forms a 45-degree angle with the positive x-axis (since the normal is at a –45-degree angle to the horizontal).
The equation of a line in standard form is Ax + By = C. For a line that makes a 45-degree angle with the positive x-axis, the slope (m) will be -1 (since tangent of 45 degrees is 1, and the slope is negative due to the angle with respect to the horizontal). The equation of the line in slope-intercept form with a slope of -1 through the point of release (r cos(θ), r sin(θ)), which is (1.25, -1.25) since r = 1.25 feet and θ = 45 degrees, is y = -x + b.
To find b, which is the y-intercept, we can substitute the x and y values of the release point into the equation: -1.25 = -1.25 * 1 + b, which gives us b = 0. The equation in slope-intercept form is y = -x.
To convert this to standard form, we rewrite it as x + y = 0.