Can someone check my answers?
1. Convert 13pi/30 to degree measure.
78°
2. Find the distance between (4,4) and (8,7).
5
3. What is the vertex of the parabola y = (x + 8)^2 - 2?
(-8,-2)
4. The graph of y = 6(x - 8)^2 + 1 open downward.
False.
5. An angle that measures pi radians equals 360°.
False.
6. What is the center of the graph of 6(x + 9)^2 + 4(y - 2)^2 = 36?
(-9,2)
7. Find the four corners of the fundamental rectangle of the hyperbola: x^2/81 - y^2/36 = 1.
(±9,±6)
8. Find the exact solutions of x^2 - (y - 6)^2 = 36 and y = -x^2.
-x^4 - 11x^2 - 36?
9. If sin B = 5√2/5√3, find the value of csc B.
√6/2
10. Which value is greater: sin 60° or cos 30°?
They are equal.
Thanks in advance
(I reposted this in case no one saw my question)
all good except #8
8.
it asked for the actual solutions.
x^2 - (-x^2 - 6)^2 = 36
x^2 - (x^4 + 12x^2 + 36) = 36
x^2 - x^4 - 12x^2 - 36 = 36
x^4 + 11x^2 + 72 = 0
no real roots, the discriminant is negative.
http://www.wolframalpha.com/input/?i=plot+x%5E2+-+(y+-+6)%5E2+%3D+36+,+y+%3D+-x%5E2
To check your answers, let's go through each question one by one:
1. Convert 13pi/30 to degree measure.
To convert radians to degrees, we use the formula: degrees = (radians * 180) / pi.
So, degrees = (13pi/30) * (180/pi) = 13 * 6 = 78°.
Your answer of 78° is correct.
2. Find the distance between (4,4) and (8,7).
To find the distance between two points (x1, y1) and (x2, y2), you can use the distance formula:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Given (4,4) and (8,7):
distance = sqrt((8 - 4)^2 + (7 - 4)^2) = sqrt(4^2 + 3^2) = sqrt(16 + 9) = sqrt(25) = 5.
Your answer of 5 is correct.
3. What is the vertex of the parabola y = (x + 8)^2 - 2?
The vertex form of a parabola is y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
Comparing y = (x + 8)^2 - 2 to the vertex form, we can see that the vertex is at (-8, -2).
Your answer of (-8, -2) is correct.
4. The graph of y = 6(x - 8)^2 + 1 opens downward.
To determine the direction of opening for a parabolic function in the form y = a(x - h)^2 + k, we look at the coefficient of the squared term (a).
Since the coefficient is positive (6), the parabola opens upward, not downward.
Your answer of False is correct.
5. An angle that measures pi radians equals 360°.
False. One complete revolution or circle is equal to 2π radians or 360°. Therefore, an angle that measures π radians is only half of a circle and corresponds to 180°.
Your answer of False is correct.
6. What is the center of the graph of 6(x + 9)^2 + 4(y - 2)^2 = 36?
To find the center of the graph of an ellipse in the form (x - h)^2/a^2 + (y - k)^2/b^2 = 1, the center is given by the point (h, k).
Comparing the given equation to the standard form, we have (x + 9)^2/6^2 + (y - 2)^2/2^2 = 1.
The center of the ellipse is (-9, 2).
Your answer of (-9, 2) is correct.
7. Find the four corners of the fundamental rectangle of the hyperbola: x^2/81 - y^2/36 = 1.
The general equation of a hyperbola in standard form is (x - h)^2/a^2 - (y - k)^2/b^2 = 1.
Comparing it to the given equation x^2/81 - y^2/36 = 1, we can see that h = 0, k = 0, a = 9, and b = 6.
The corners can be found as (h ± a, k ± b), giving us (±9, ±6) as the four corners of the fundamental rectangle.
Your answer of (±9, ±6) is correct.
8. Find the exact solutions of x^2 - (y - 6)^2 = 36 and y = -x^2.
To find the exact solutions, we need to solve the system of equations x^2 - (y - 6)^2 = 36 and y = -x^2 simultaneously.
Substituting y = -x^2 into the first equation, we have x^2 - (-x^2 - 6)^2 = 36.
Simplifying the expression, we get x^4 + 11x^2 + 36 = 0.
The exact solutions for this equation are not readily apparent, so it seems like you may have made a mistake in your answer. To find the exact solutions, you may need to use numerical methods or factoring techniques.
9. If sin B = 5√2/5√3, find the value of csc B.
The reciprocal of sine is cosecant (csc). Therefore, to find the value of csc B, we take the reciprocal of sin B.
csc B = 1 / sin B = 1 / (5√2/5√3).
Simplifying this, we multiply the numerator and denominator by the conjugate of the denominator (√3/√3) to rationalize the denominator.
csc B = 1 / (5√2/5√3) = (1 * √3) / (5√2 * √3) = √3 / (5√6) = √(3/6) = √(1/2) = √(1) / √(2) = 1/√2 = √2/2.
Your answer of √6/2 is incorrect. The correct answer is √2/2.
10. Which value is greater: sin 60° or cos 30°?
To compare the values of sine and cosine, we refer to the unit circle.
sin 60° is equal to √3/2, and cos 30° is equal to √3/2.
Since the values are equal, neither is greater.
Your answer of They are equal is correct.