A pair of tangents to a circle which is inclined to each other at an angle of 60 degree are drawn at ends of two radii. The angl
1 answer:
Answer:
The angle between these radii must be 120º.
Step-by-step explanation:
According to Euclidean Geometry, two tangents to a circle are symmetrical to each other and the axis of symmetry passes through the center of the circle and, hence, each tangent is perpendicular to a respective radius. We represent the statement in the diagram included below.
Then, we calculate the angle of the radius with respect to the axis of symmetry by knowing the fact that sum of internal angles within triangle equals 180º. That is to say:
And the angle between these two radii is twice the result.
The angle between these radii must be 120º.
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Answer: (a) Interval where f is increasing: (0.78,+∞);
Interval where f is decreasing: (0,0.78);
(b) Local minimum: (0.78, - 0.09)
(c) Inflection point: (0.56,-0.06)
Interval concave up: (0.56,+∞)
Interval concave down: (0,0.56)
Step-by-step explanation:
(a) To determine the interval where function f is increasing or decreasing, first derive the function:
f'(x) = [
]
Using the product rule of derivative, which is: [u(x).v(x)]' = u'(x)v(x) + u(x).v'(x),
you have:
f'(x) =
f'(x) =
f'(x) =
Now, find the critical points: f'(x) = 0
= 0
x = 0
and
x =
x = 0.78
To determine the interval where f(x) is positive (increasing) or negative (decreasing), evaluate the function at each interval:
interval x-value f'(x) result
0<x<0.78 0.5 f'(0.5) = -0.22 decreasing
x>0.78 1 f'(1) = 1 increasing
With the table, it can be concluded that in the interval (0,0.78) the function is decreasing while in the interval (0.78, +∞), f is increasing.
Note: As it is a natural logarithm function, there are no negative x-values.
(b) A extremum point (maximum or minimum) is found where f is defined and f' changes signs. In this case:
- Between 0 and 0.78, the function decreases and at point and it is defined at point 0.78;
- After 0.78, it increase (has a change of sign) and f is also defined;
Then, x=0.78 is a point of minimum and its y-value is:
f(x) =
f(0.78) =
f(0.78) = - 0.092
The point of <u>minimum</u> is (0.78, - 0.092)
(c) To determine the inflection point (IP), calculate the second derivative of the function and solve for x:
f"(x) = [
]
f"(x) =
f"(x) =
= 0
and
Substituing x in the function:
f(x) =
f(0.56) =
f(0.56) = - 0.06
The <u>inflection point</u> will be: (0.56, - 0.06)
In a function, the concave is down when f"(x) < 0 and up when f"(x) > 0, adn knowing that the critical points for that derivative are 0 and 0.56:
f"(x) =
f"(0.1) =
f"(0.1) = - 0.21, i.e. <u>Concave</u> is <u>DOWN.</u>
f"(0.7) =
f"(0.7) = + 1.33, i.e. <u>Concave</u> is <u>UP.</u>
Jenny won the race as she finished the race much more quickly and has greater average speed than Lester.
Jenny: distance/time = 100/15.2 = 6.58 m/s = 6.6 m/s
Lester: distance/time = 100/15.7 = 6.37 m/s = 6.4 m/s
Data:
l and m are parallel lines
QR and ST are perpendicular in R
Angle 1 is 63°
The angle formed by perpendicular lines is a right angle (90°)
Angles 1 and 3 are alternate angles: angles that occur on opposite sides of a transversal line that is crossing two parallel.
Alternate angles are congruent, have the same measure.
The sum of the interior angles of a triangle is always 180°. In triangle QRT:
Use the equation above to find the measure of angle 2:
