Line AB has a midpoint (2,5). A has the coordinates (1,7). find the coordinates of B.
1 answer:
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Assuming ballistic motion, the equation for height as a function of time is
h(t) = -16t² + v0·t + h0
where v0 is the initial vertical velocity (44 ft/s) and h0 is the initial height (8 ft).
You want to find t such that h(t) = 15. Substituting the given values, we have
15 = -16t² +44t +8
16t² -44t +7 = 0
Using the quadratic formula, we find t to be ...
t = (44 ±√(44² - 4·16·7))/(2·16)
t = (11 ± √93)/8
We are not interested in the time when the football is rising through the 15 ft height, so the time of interest is
t = (11 +√93)/8 ≈ 2.580 . . . . seconds
h(t) = -16t² + v0·t + h0
where v0 is the initial vertical velocity (44 ft/s) and h0 is the initial height (8 ft).
You want to find t such that h(t) = 15. Substituting the given values, we have
15 = -16t² +44t +8
16t² -44t +7 = 0
Using the quadratic formula, we find t to be ...
t = (44 ±√(44² - 4·16·7))/(2·16)
t = (11 ± √93)/8
We are not interested in the time when the football is rising through the 15 ft height, so the time of interest is
t = (11 +√93)/8 ≈ 2.580 . . . . seconds
Answer:
50π ≈ 157.08 cubic units
Step-by-step explanation:
The volume of revolution of a plane figure is the product of the area of the figure and the length of the path of revolution of the centroid of that area. The centroid of a triangle is 1/3 the distance from each side to the opposite vertex. (It is the intersection of medians.)
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<h3>length of centroid path</h3>One side of this triangle is the axis of revolution. Then the radius to the centroid is 1/3 the x-dimension of the triangle, so is 5/3. Then the circumference of the circle along which the centroid is revolved is ...
C = 2πr
C = 2π(5/3) = 10π/3 . . . units
<h3>triangle area</h3>The area of the triangle is found using the formula ...
A = 1/2bh
A = 1/2(5)(6) = 15 . . . square units
<h3>volume</h3>The volume is the product of the area and the path length:
V = AC
V = (15)(10π/3) = 50π . . . cubic units
The volume of the solid is 50π ≈ 157.08 cubic units.
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<em>Additional comment</em>
In the attached figure, the point labeled D is the centroid of the triangle. The label has no significance other than being the next after A, B, C were used to label the vertices.
The volume of revolution can also be found using integration and "shell" or "disc" differential volumes. The result is the same.
