These are so great! They are a perfect combination of Physics and pre-calculus! Your max height of that projectile is going to occur at the max value of the parabola, or at its vertex. So we need to find the vertex. The coordinates of the vertex will give us the x value, which is the time in seconds it takes to reach y which is the max height. Do this by completing the square. Begin by setting the equation equal to 0 and then moving the 80 over to the other side. Then factor out the -16. This is all that: 
. Take half the linear term which is 4 and square it and add it in to both sides. Half of 4 is 2, 2 squared is 4, so add 4 into the set of parenthesis and to the -80. 
. The -64 on the right comes from the fact that when you added 4 into the parenthesis, you had the -16 out in front which is a multiplier. -16 * 4 - -64. So what you really added in was -64. Now the perfect square binomial we created in that process was 
. When we move the 144 back over by addition we find that the vertex of the polynomial is (2, 144). And that tells us that it takes 2 seconds for the projectile to reach its max height of 144 feet. To find the time interval in which the object's height decreases occurs from its max height of 144 to where the graph of the parabola goes through the x-axis to the right of the max. To find where the graph goes through the x-axis, or the zeroes of the graph, you factor the polynomial. When you do that using the quadratic formula you get that x = -1 and 5. So at its max height it is at 2 seconds, and by 5 seconds it hits the ground. So the time interval of its height decreasing is from 2 seconds to 5 seconds, or a total of 3 seconds. I think you need the 2 and 5, from the wording of your problem.
Find the length of the missing side of the triangle below PLEASE HELPP!!
1 answer:
To find the length of the missing side, you can use the Pythagorean Theorem, which you can only use for right triangles.
Pythagorean Theorem: a² + b² = c²
[ c is the hypotenuse or the longest side of the triangle, and a and b are the other sides, I think it doesn't matter which side ]
Plug in what you know:
b = 8
c = 14
a² + b² = c²
a² + (8)² = (14)²
a² + 64 = 196 Subtract 64 on both sides
a² = 132 Square root both sides to get "a" by itself
a = inches
If you need to simplify more, you need to find two numbers that multiply to = 132 where one of them can be square rooted, to do so you can list the greatest common factors of 132
132: 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132
The only number that can be square rooted is 4, so you can do:
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The area is maximized when half the fence is used along the river. The length and width of the plot are 36 ft and 18 ft.
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Let x represent the length of fence parallel to the river. Then the area (A) can be described by
.. A = x(72 -x)/2
This equation describes a downward-opening parabola with zeros at x=0 and x=72. Its vertex (maximum) is halfway between the zeros, at x=36.
_____
Let x represent the length of fence parallel to the river. Then the area (A) can be described by
.. A = x(72 -x)/2
This equation describes a downward-opening parabola with zeros at x=0 and x=72. Its vertex (maximum) is halfway between the zeros, at x=36.