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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Answer:
x = ± - 3
Explanation:
I'm assuming you want the solutions to that equation, so here goes! (If not, please comment.)
(x-3)(x+9)=27
Let's FOIL this all out and expand. (Remember: First, Outer, Inner, Last.)
x^2 + 9x - 3x - 27
(first+ inner + outer + last)
x^2 + 9x - 3x - 27 = 27
Combine like terms, and add 27 to both sides.
x^2 + 6x - 27 + 27 = 27 + 27
x^2 + 6x = 54
Let's complete the square, because factoring doesn't work, and because it's good practice.
x^2 + 6x + ___ = 54 + ____
In the blank we will put b/2 ^2 = 6/2 ^2 = 3^2 = 9 to complete the square.
x^2 + 6x + 9 = 54 + 9
Now we've got a perfect square factor:
(x + 3)^2 = 63
sqrt(x+3)^2 =
x + 3 = ±
x = ± - 3