A woodland jumping mouse hops along a parabolic path given by y=-0.2x^2+1.3x where x is the mouse's horizontal position (in feet) and y is the corresponding height (in feet). Can the mouse jump over a fence 3 feet high? Explain.
I know that x is 3.25 and the mouse's horizontal position in feet and I know that y is 2.1 (2.1125) and the corresponding height in feet. I don't understand how to tell if the mouse can jump over a fence that is three feet high. Could someone please explain this to me?
lets look at the graph.
You can plot the graph on your calculator, and see the max height of the mouse. If it is over 3, it can.
with calculus
finding the max
dy/dx=0=-.4x+1.3
x=1.3/.4=3.25
ymax=-.2(3.25^2)+1.3(3.25)=2.11 The mouse will smash into the fence.
What does the horizontal position mean?
Trains travel horizonally, rockets start off mostly vertical (up)
Horizonal means no elevation change.
So that's how far it has to jump and y is how high it can jump from that point to the fence?
To determine if the mouse can jump over a fence that is three feet high, we need to compare the maximum height reached by the mouse to the height of the fence.
The equation given is y = -0.2x^2 + 1.3x, where x is the horizontal position of the mouse and y is the corresponding height. To find the maximum height, we need to find the vertex (h, k) of the parabolic path. The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a and b are coefficients from the equation in the form y = ax^2 + bx + c.
In this case, a = -0.2 and b = 1.3. Plugging these values into the formula, we get:
x = -1.3 / (2 * -0.2)
x = -1.3 / -0.4
x = 3.25
So the x-coordinate of the vertex is 3.25. To find the corresponding y-coordinate, we can substitute this value back into the equation:
y = -0.2(3.25)^2 + 1.3(3.25)
y = -0.2(10.5625) + 4.225
y = -2.1125 + 4.225
y = 2.1125
The maximum height reached by the mouse is 2.1125 feet.
Now, let's compare this height with the height of the fence, which is 3 feet. Since the maximum height (2.1125 feet) is less than the height of the fence (3 feet), the mouse cannot jump over the fence.
In summary, the mouse cannot jump over a fence that is three feet high, as its maximum height is only 2.1125 feet.