Show your work
1. write an equation of an ellipse in standard form with the center at the orgin and with the given characteristics.
Vertix (-5,0) and co-vertex at (0, 4)
A. x^2/25 + y^2/16 = 1
B. x^2/4 + y^2/5= 1
To write the equation of an ellipse in standard form with the center at the origin, we need to use the following equation:
x^2/a^2 + y^2/b^2 = 1
where (0, 0) is the center of the ellipse, a is the length of the semi-major axis (half the length of the longer axis), and b is the length of the semi-minor axis (half the length of the shorter axis).
From the given characteristics, we know that the vertex is (-5,0) which means that a = 5 (since the distance from the center to a vertex is the length of the semi-major axis). We also know that the co-vertex is (0,4) which means that b = 4 (since the distance from the center to a co-vertex is the length of the semi-minor axis).
Using these values, we can plug them into the standard form equation and simplify:
x^2/5^2 + y^2/4^2 = 1
x^2/25 + y^2/16 = 1
Therefore, the equation of the ellipse in standard form with the center at the origin and with the given characteristics is A. x^2/25 + y^2/16 = 1.
Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
Participants in a study of a new medication received either medication A or a placebo. Find P(placebo and improvement). You may find it helpful to make a tree diagram of the problem on a separate piece of paper.
Of all those who participated in the study, 80% received medication A.
Of those who received medication A, 76% reported an improvement.
Of those who received the placebo, 62% reported no improvement.
To find P(placebo and improvement), we need to use the formula:
P(placebo and improvement) = P(placebo) * P(improvement | placebo)
From the given information, we know that:
- P(placebo) = 1 - P(medication A) = 1 - 0.8 = 0.2
- P(no improvement | placebo) = 0.62 (since 62% reported no improvement, the complement, or those who reported improvement, is 1 - 0.62 = 0.38)
- P(improvement | placebo) = 1 - P(no improvement | placebo) = 1 - 0.62 = 0.38
Therefore, we can plug in these values to the formula:
P(placebo and improvement) = 0.2 x 0.38 = 0.076
So P(placebo and improvement) is 0.076, or 7.6%.
To write the equation of an ellipse in standard form with the center at the origin and given characteristics, we need to know the lengths of the major and minor axes.
The vertices of the ellipse give us the lengths of the major axis. In this case, the distance between the vertices is 2 times the value of "a". Therefore, "a" = |-5 - 0| / 2 = 5/2.
The co-vertices give us the lengths of the minor axis. In this case, the distance between the co-vertices is 2 times the value of "b". Therefore, "b" = |0 - 4| / 2 = 2.
The equation of an ellipse in standard form is:
(x^2) / (a^2) + (y^2) / (b^2) = 1.
Plugging in the values for "a" and "b", we get:
(x^2) / (25/4) + (y^2) / 4 = 1.
Multiplying both sides of the equation by 25/4, we can simplify further:
(x^2) / 25 + (y^2) / 16 = 1.
Therefore, the equation of the ellipse in standard form with the given characteristics is:
A. x^2/25 + y^2/16 = 1.
To find the equation of an ellipse with the center at the origin, we need to determine the values for the major and minor axes. The major axis is twice the distance between the center and the vertex, and the minor axis is twice the distance between the center and the co-vertex.
Given that the vertex is (-5, 0), the distance from the center to the vertex is 5. So the major axis is 2(5) = 10.
Given that the co-vertex is (0, 4), the distance from the center to the co-vertex is 4. So the minor axis is 2(4) = 8.
Now, we can use these values to write the equation of the ellipse in standard form, which is (x^2/a^2) + (y^2/b^2) = 1, where "a" represents the semi-major axis and "b" represents the semi-minor axis.
In this case, the semi-major axis is 10/2 = 5, and the semi-minor axis is 8/2 = 4.
Therefore, the equation of the ellipse is (x^2/5^2) + (y^2/4^2) = 1.
Simplifying, we get x^2/25 + y^2/16 = 1.
So the correct answer is option A: x^2/25 + y^2/16 = 1.