5. Shara's new lawn sprinkler watered a circular area with three times the radius of the old sprinkler. How many times greater was the area watered by the new sprinkler than the area watered by the old sprinkler?
a)3
b)6
c)9
d)36
6.How many x-intercepts does the graph of the equation below have ?
y = 2x^2 - x - 3
a)3
b)2
c)1
d)0
My answers: (please check)
5. A
6. A
5 ok
6 huh? How can a parabola have three roots? Did you try graphing it? Remember the quadratic formula?
recall that an nth degree polynomial has at most n real roots
oops. #5 the area is 3^2 times as big.
5. C
6. B
much better
your screen name reminds me of this one:
aibohphobia: fear of palindromes
To find the answers to these questions, let's break down each problem step by step.
5. Shara's new lawn sprinkler watered a circular area with three times the radius of the old sprinkler. To compare the areas watered by the two sprinklers, we need to find the ratio of the areas.
The formula for the area of a circle is A = π * r^2, where A is the area and r is the radius.
Let's assume the radius of the old sprinkler is r. Then the radius of the new sprinkler is 3r.
The area of the old sprinkler is A_old = π * r^2.
The area of the new sprinkler is A_new = π * (3r)^2 = π * 9r^2.
To find the ratio of the areas, we divide the area of the new sprinkler by the area of the old sprinkler:
A_new / A_old = (π * 9r^2) / (π * r^2) = 9.
So the area watered by the new sprinkler is 9 times greater than the area watered by the old sprinkler, which means the correct answer is option c) 9.
6. To find the x-intercepts of the graph of the equation y = 2x^2 - x - 3, we need to set y = 0 and solve for x.
Let's set y = 0:
0 = 2x^2 - x - 3.
To solve this quadratic equation, we can factor it or use the quadratic formula. In this case, factoring doesn't seem straightforward, so let's use the quadratic formula:
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a).
In our equation, a = 2, b = -1, and c = -3. Plugging these values into the quadratic formula, we can solve for x:
x = (-(-1) ± √((-1)^2 - 4(2)(-3))) / (2(2))
= (1 ± √(1 + 24)) / 4
= (1 ± √25) / 4
= (1 ± 5) / 4.
This gives us two possible solutions:
x = (1 + 5) / 4 = 6 / 4 = 3/2, and
x = (1 - 5) / 4 = -4 / 4 = -1.
So the equation has two x-intercepts, which means the correct answer is option b) 2.