Question

a toy rocket is launched at an initial velocity of 50 ft/s at an angle of 75° with the horizontal. how long will it take for the rocket to travel 20 feet horizontally?
2.6 seconds
1.5 seconds
9.7 seconds
0.4 second
question 17 (5 points)
identify the conic given by \\( \frac { ( x - 8 ) ^ { 2 } } { 64 } - \frac { ( y - 9 ) ^ { 2 } } { 4 } = 1 \\)
hyperbola
circle
ellipse
parabola

Explanation:

Step1: Find the horizontal component of velocity

The horizontal component of velocity \(v_x\) is given by \(v_x = v_0\cos\theta\), where \(v_0 = 50\) ft/s and \(\theta=75^{\circ}\).
\(v_x=50\cos75^{\circ}\approx50\times0.2588 = 12.94\) ft/s.

Step2: Use the formula for horizontal motion

The formula for horizontal motion is \(x = v_x t\), where \(x = 20\) ft.
We need to solve for \(t\), so \(t=\frac{x}{v_x}\).
Substitute \(x = 20\) and \(v_x\approx12.94\) into the formula: \(t=\frac{20}{12.94}\approx1.5\) s.

For the conic \(\frac{(x - 8)^2}{64}-\frac{(y - 9)^2}{4}=1\), the general form of a hyperbola is \(\frac{(x - h)^2}{a^2}-\frac{(y - k)^2}{b^2}=1\) (for a hyperbola opening left - right) or \(\frac{(y - k)^2}{a^2}-\frac{(x - h)^2}{b^2}=1\) (for a hyperbola opening up - down). Here \(h = 8,k = 9,a^2=64,b^2 = 4\), and the equation is of the form \(\frac{(x - h)^2}{a^2}-\frac{(y - k)^2}{b^2}=1\). A circle has the form \((x - h)^2+(y - k)^2=r^2\), an ellipse has the form \(\frac{(x - h)^2}{a^2}+\frac{(y - k)^2}{b^2}=1\) (\(a
eq b\)) or \(\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1\) (\(a
eq b\)), and a parabola has the form \(y=ax^{2}+bx + c\) (vertical parabola) or \(x = ay^{2}+by + c\) (horizontal parabola).

Answer:

1.5 seconds