What is the area of the figure?10.2 in.3 in.3 in2 in.2 in.8 in.6.2 in
1 answer:
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Option C:
x = 30
Solution:
The given image is a triangle.
angle 1, angle 2 and angle 3 are interior angles of a triangle.
angle 4 is the exterior angle of a triangle.
m∠4 = 2x°, , m∠3 = 20°
Exterior angle theorem:
<em>In triangle, the measure of exterior angle is equal to the sum of the opposite interior angles.</em>
By this theorem,
m∠4 = m∠2 + m∠3
Subtract on both sides of the equation.
To make the denominator same and then subtract.
Multiply by on both sides of the equation.
x° = 30°
x = 30
Hence option C is the correct answer.
1) The function is 3(x + 2)³ - 3
2) The end behaviour is the limits when x approaches +/- infinity.
3) Since the polynomial is of odd degree you can predict that the ends head off in opposite direction. The limits confirm that.
4) The limit when x approaches negative infinity is negative infinity, then the left end of the function heads off downward (toward - ∞).
5) The limit when x approaches positive infinity is positivie infinity, then the right end of the function heads off upward (toward + ∞).
6) To graph the function it is important to determine:
- x-intercepts
- y-intercepts
- critical points: local maxima, local minima, and inflection points.
7) x-intercepts ⇒ y = 0
⇒ <span>3(x + 2)³ - 3 = 0 ⇒ (x + 2)³ - 1 = 0
</span>
<span>⇒ (x + 2)³ = -1 ⇒ x + 2 = 1 ⇒ x = - 1
</span>
8) y-intercepts ⇒ x = 0
y = <span>3(x + 2)³ - 3 = 3(0 + 2)³ - 3 = 0 - 3×8 - 3 = 24 - 3 = 21
</span><span>
</span><span>
</span><span>9) Critical points ⇒ first derivative = 0
</span><span>
</span><span>
</span><span>i) dy / dx = 9(x + 2)² = 0
</span><span>
</span><span>
</span><span>⇒ x + 2 = 0 ⇒ x = - 2
</span><span>
</span><span>
</span><span>ii) second derivative: to determine where x = - 2 is a local maximum, a local minimum, or an inflection point.
</span><span>
</span><span>
</span><span>y'' = 18 (x + 2); x = - 2 ⇒ y'' = 0 ⇒ inflection point.
</span><span>
</span><span>
</span><span>Then the function does not have local minimum nor maximum, but an inflection point at x = -2.
</span><span>
</span><span>
</span><span>Using all that information you can graph the function, and I attache the figure with the graph.
</span>
2) The end behaviour is the limits when x approaches +/- infinity.
3) Since the polynomial is of odd degree you can predict that the ends head off in opposite direction. The limits confirm that.
4) The limit when x approaches negative infinity is negative infinity, then the left end of the function heads off downward (toward - ∞).
5) The limit when x approaches positive infinity is positivie infinity, then the right end of the function heads off upward (toward + ∞).
6) To graph the function it is important to determine:
- x-intercepts
- y-intercepts
- critical points: local maxima, local minima, and inflection points.
7) x-intercepts ⇒ y = 0
⇒ <span>3(x + 2)³ - 3 = 0 ⇒ (x + 2)³ - 1 = 0
</span>
<span>⇒ (x + 2)³ = -1 ⇒ x + 2 = 1 ⇒ x = - 1
</span>
8) y-intercepts ⇒ x = 0
y = <span>3(x + 2)³ - 3 = 3(0 + 2)³ - 3 = 0 - 3×8 - 3 = 24 - 3 = 21
</span><span>
</span><span>
</span><span>9) Critical points ⇒ first derivative = 0
</span><span>
</span><span>
</span><span>i) dy / dx = 9(x + 2)² = 0
</span><span>
</span><span>
</span><span>⇒ x + 2 = 0 ⇒ x = - 2
</span><span>
</span><span>
</span><span>ii) second derivative: to determine where x = - 2 is a local maximum, a local minimum, or an inflection point.
</span><span>
</span><span>
</span><span>y'' = 18 (x + 2); x = - 2 ⇒ y'' = 0 ⇒ inflection point.
</span><span>
</span><span>
</span><span>Then the function does not have local minimum nor maximum, but an inflection point at x = -2.
</span><span>
</span><span>
</span><span>Using all that information you can graph the function, and I attache the figure with the graph.
</span>