Problem 1
Answer: Choice A
(30, 225); Rocket reaches max height of 225 ft after 30 seconds.
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Explanation:
Distribute the 0.25 through to get
y = 0.25(60x-x^2)
y = 15x - 0.25x^2
y = -0.25x^2 + 15x
This is in standard form y = ax^2+bx+c with a = -0.25, b = 15 and c = 0.
The first two values mentioned lead to this x coordinate of the vertex.
h = -b/(2a)
h = -15/(2*(-0.25))
h = -15/(-0.5)
h = 30
This tells us the answer is between A and C.
If you were to plug x = 30 into the original equation, you'll get y = 225. It means that after x = 30 seconds, the rocket has reached its max height of 225 feet. Therefore, the answer must be choice A.
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Problem 2
Answer: Choice B
(x-3)^2 = -8(y-6)
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Explanation:
vertex = base of the bulb = (3,6)
focus = top of the bulb = (3,4)
This flashlight is pointed downward, which forms a downward opening parabola.
The distance from the vertex to the focus is p = 2 units. This is known as the focal distance.
We'll plug that in along with the vertex (h,k) = (3,6) in the formula below
4p(y-k) = (x-h)^2
4*2(y-6) = (x-3)^2
8(y-6) = (x-3)^2
Unfortunately that equation above produces a parabola that opens upward. To fix things, we stick a negative out front on either side, which will reflect the parabola over the x axis to make the parabola open downward. That leads us to choice B.
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Problem 3
Answer: D) square
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Explanation:
We can form a point by intersecting the plane through the point where the two cones meet. The plane needs to be parallel to the base of each cone.
We can also form a line. To do so, we intersect the plane at exactly along the edge of the cone. Make the plane tangent to the cone.
Lastly, we can form a circle by intersecting any plane parallel to the cone's base. This plane cannot pass through the meeting point of the two cones. Rather, the plane passes through one of the cones.
Those three previous paragraphs mean that we can rule out choices A,B,C. The only thing we can't form is a square. We can form straight lines, but we cannot form perpendicular lines needed to get a square. Nice work on selecting the correct answer.
