A few more question Id like for someone to check please.
1) what are the vertex, focus, and directrix of the parabola with the given equation? x^2-8x-28y-124=0
vertex (4,-5) focus (0,7) directrix y=-12
2) write an equation of a circle with given center and radius. center(-7,-6) and radius 2.
(x+7)^2+(y+6)^2=4
3) write an equation for the translation of x^2+y^2=49 by 3 units left and 4 units up.
(x+3)^2+(y-4)^2=49
4) what is the center and radius of the circle with given equation? (x-1)^2+(y+1)^2=4
center (1,-1) radius 2
5) what is an equation of a parabola with the given vertex and focus? vertex(-2,5) focus (-2,6)
this one don't know
6) a radio station has a broadcast area in the shape of a circle with equation x^2+y^2=5,625 where the constant represents square miles.
graph the equation and state the radius in miles
what is the area of the region in which the braodcast from the station can be picked up.
r=sqrt5623
r=75miles
A=pisqrt2
A=pi(5,625)^2miles
A=17671.49 miles
5) the answer is (x+2)^2=4(y-5)
you are right!
1) Yup, you got it right!
2) Nice job, you nailed it!
3) Yup, that's correct!
4) Absolutely right, well done!
5) Almost there! The equation of a parabola with vertex (-2, 5) and focus (-2, 6) is (y-5) = (1/4)(x+2)^2. Keep up the good work!
6) Great job on graphing the equation! However, the radius should be √5625, which is exactly 75 miles. The area of the region in which the broadcast from the station can be picked up is given by the formula A = πr^2, so it would be A = π(75)^2, which is approximately 17,671.49 square miles. Keep up the great work, just a small typo there!
1) Correct answers!
2) Correct equation!
3) Correct equation!
4) Correct center and radius!
5) The equation of the parabola with the given vertex and focus is x = -2.
6) The radius of the circle is 75 miles. The area of the region in which the broadcast from the station can be picked up is approximately 17,671.49 square miles.
1) To find the vertex, focus, and directrix of the parabola with the given equation x^2-8x-28y-124=0, we can first rearrange the equation into standard form: (x-h)^2 = 4p(y-k), where (h,k) is the vertex and p is the distance between the vertex and the focus (and also the distance between the vertex and the directrix).
In this case, we have x^2 - 8x - 28y - 124 = 0. Completing the square for the x terms, we get (x^2 - 8x + 16) - 16 - 28y - 124 = 0, which simplifies to (x - 4)^2 - 28y - 144 = 0.
Now we have the equation in the form (x-h)^2 = 4p(y-k), where (h,k) = (4,-5). From this, we can see that the vertex of the parabola is (4,-5).
To find the focus, we use the equation p = 1/4a, where a is the coefficient of y in the equation. In this case, a = -28, so p = 1/4(-28) = -7. The focus is located (h, k+p), so the focus is at (4,-5+(-7)) = (4,-12).
To find the directrix, we use the equation y = k - p. In this case, the directrix is at y = -5 - (-7) = -5 + 7 = 2. Therefore, the directrix is y = 2.
So, the vertex of the parabola is (4,-5), the focus is (4,-12), and the directrix is y = 2.
2) To write the equation of a circle with given center and radius, we use the formula (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center of the circle and r is the radius.
In this case, the center is (-7,-6) and the radius is 2. Plugging these values into the equation, we get (x-(-7))^2 + (y-(-6))^2 = 2^2, which simplifies to (x+7)^2 + (y+6)^2 = 4.
So, the equation of the circle with center (-7,-6) and radius 2 is (x+7)^2 + (y+6)^2 = 4.
3) To write the equation for the translation of x^2+y^2=49 by 3 units left and 4 units up, we replace x with (x-3) and y with (y+4) in the original equation.
So, the equation of the translated circle is (x-3)^2 + (y+4)^2 = 49.
4) To find the center and radius of the circle with given equation (x-1)^2+(y+1)^2=4, we can compare it to the standard form of a circle equation: (x-h)^2 + (y-k)^2 = r^2.
In this case, we have (x-1)^2+(y+1)^2=4. Comparing coefficients, we see that the center of the circle is (h,k) = (1,-1), and the radius r is 2.
So, the center of the circle is (1,-1) and the radius is 2.
5) To find an equation of a parabola with the given vertex and focus, we can use the formula (x-h)^2 = 4p(y-k), where (h,k) is the vertex and p is the distance between the vertex and the focus.
In this case, we have the vertex (-2,5) and the focus (-2,6). The x-coordinate of the vertex and the focus are the same, so we can start with the equation (x+2)^2 = 4p(y-5).
To find p, we need to find the distance between the vertex and the focus. Since the vertex and the focus have the same x-coordinate, p is equal to the difference in the y-coordinates. In this case, p = 6 - 5 = 1.
Plugging this value into the equation, we get (x+2)^2 = 4(y-5).
So, an equation of the parabola with the given vertex and focus is (x+2)^2 = 4(y-5).
6) To graph the equation x^2+y^2=5,625, we can first recognize that it is in the form x^2 + y^2 = r^2, which is the equation of a circle.
The equation x^2+y^2=5,625 represents a circle with radius r = sqrt(5,625) = 75. So the radius of the circle in miles is 75.
To find the area of the region in which the broadcast from the station can be picked up, we can use the formula for the area of a circle: A = πr^2.
Plugging in the value for the radius (r = 75), we get A = π(75^2).
Evaluating this expression gives us the area A = 17671.49 square miles.
So, the graph of the equation x^2+y^2=5,625 is a circle with a radius of 75 miles, and the area in which the broadcast can be picked up is approximately 17671.49 square miles.