An object is moving around the unit circle with parametric equations x(t)=cos(t), y(t)=sin(t), so it's location at time t is P(t
)=(cos(t),sin(t)) . Assume 0 < t < π/2. At a given time t, the tangent line to the unit circle at the position P(t) will determine a right triangle in the first quadrant. (Connect the origin with the y-intercept and x-intercept of the tangent line.)(a) The area of the right triangle is a(t)= 1/(2sin(x)cos(x)
(b)
lim
t ? pi/2?
a(t)= +infinity
(c)
lim
t ? 0+
a(t)= -infinity
(d)
lim
t ? pi/4
a(t)= 1
(e) With our restriction on t, the smallest t so that a(t)=2 is ??
(f) With our restriction on t, the largest t so that a(t)=2 is ??
(g) The average rate of change of the area of the triangle on the time interval [?/6,?/4] is ??
(h) The average rate of change of the area of the triangle on the time interval [?/4,?/3] is ??
(i) Create a table of values to study the average rate of change of the area of the triangle on the time intervals [?/6,b], as b approaches ?/6 from the right. The limiting value is -4/3
(j) Create a table of values to study the average rate of change of the area of the triangle on the time intervals [a,?/3], as a approaches ?/3 from the left. The limiting value is 4/3
(b)
lim
t ? pi/2?
a(t)= +infinity
(c)
lim
t ? 0+
a(t)= -infinity
(d)
lim
t ? pi/4
a(t)= 1
(e) With our restriction on t, the smallest t so that a(t)=2 is ??
(f) With our restriction on t, the largest t so that a(t)=2 is ??
(g) The average rate of change of the area of the triangle on the time interval [?/6,?/4] is ??
(h) The average rate of change of the area of the triangle on the time interval [?/4,?/3] is ??
(i) Create a table of values to study the average rate of change of the area of the triangle on the time intervals [?/6,b], as b approaches ?/6 from the right. The limiting value is -4/3
(j) Create a table of values to study the average rate of change of the area of the triangle on the time intervals [a,?/3], as a approaches ?/3 from the left. The limiting value is 4/3
1 answer:
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Given:
The width at the base of parabolic tunnel is 7 m.
The ceiling 3 m from each end of the base there are light fixtures.
The height to light fixtures is 4 m.
To find:
Whether it is possible a trailer truck carrying cars is 4 m wide and 2.8 m high is going to drive through the tunnel.
Solution:
The width at the base of tunnel is 7 m.
Let the graph of the parabola intersect the x-axis at x=0 and x=7. It means x and (x-7) are the factors of the height function.
The function of height is:
...(i)
Where, a is a constant.
The ceiling 3 m from each end of the base there are light fixtures and the height to light fixtures is 4 m. It means the graph of height function passes through the point (3,4).
Putting x=3 and h(x)=4 in (i), we get
Putting , we get
...(ii)
The center of the parabola is the midpoint of 0 and 7, i.e., 3.
The width of the truck is 4 m. If is passes through the center then the truck must m 2 m on the left side of the center and 2 m on the right side of the center.
2 m on the left side of the center is x=1.5.
A trailer truck carrying cars is 4 m wide and 2.8 m high is going to drive through the tunnel is possible if h(1.5) is greater than 2.8.
Putting x=1.5 in (ii), we get
Since h(1.5)<2.8, therefore the trailer truck carrying cars is 4 m wide and 2.8 m high is going to drive through the tunnel is not possible.
