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dimulka [17.4K]
3 years ago
5

PLEASE HELO ME AND EXPLAIN PLEASE. AND PLEASE SHOW IN STEPS!! i will mark brainliest if you get it correct!

Mathematics
1 answer:
Dimas [21]3 years ago
4 0

Step-by-step explanation:

The height of the swipe card is given by :

h(t)=-3t^2+15t+18 .....(1)

We need to find the maximum height of the swipe card and the time to reach the maximum height.

For maximum height, put dh/dt = 0

So,

\dfrac{d}{dt}(-3t^2+15t+18)=0\\\\-9t+15=0\\\\t=\dfrac{15}{9}\\\\t=1.67\ s

It will take 1.67 seconds to reach the maximum height.

Now, put t = 1.67 s in equation (1).

h(1.67)=-3(1.67)^2+15(1.67)+18\\\\=34.68\ feet

Hence, the maximum height of the swipe card is 34.68 feet and the time to reach this height is 1.67 s.

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Derive the equation of the parabola with a focus at (−5, 5) and a directrix of y = −1.
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Answer:

The equation of the parabola with a focus at (-5,5) and a directrix of y = -1 is y = \frac{1}{12}\cdot (x+5)^{2}+2.

Step-by-step explanation:

From statement we understand that parabola has its axis of symmetry in an axis parallel to y-axis. According to Analytical Geometry, the minimum distance between focus and directrix equals to twice the distance between vertex and any of endpoints.

If endpoints are (-5, 5) and (-5, -1), respectively, then such distance (r), dimensionless, is calculated by means of the Pythagorean Theorem:

r = \frac{1}{2}\cdot \sqrt{[-5-(-5)]^{2}+[5-(-1)]^{2}}

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And the location of the vertex (V(x,y)), dimensionless, which is below the focus, is:

V(x,y) = F(x,y)-R(x,y) (1)

Where:

F(x,y) - Focus, dimensionless.

R(x,y) - Vector distance, dimensionless.

If we know that F(x,y) = (-5,5) and R(x,y) = (0,3), then the location of the vertex is:

V(x,y) = (-5,5)-(0,3)

V(x,y) =(-5,2)

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y-k = \frac{(x-h)^{2}}{4\cdot r} (2)

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h, k - Coordinates of the vertex, dimensionless.

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If we know that h = -5, k = 2 and r = 3, then the equation of the parabola is:

y = \frac{1}{12}\cdot (x+5)^{2}+2

The equation of the parabola with a focus at (-5,5) and a directrix of y = -1 is y = \frac{1}{12}\cdot (x+5)^{2}+2.

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