The graph below shows the height of a tunnel f(x), in feet, depending on the distance from one side of the tunnel x, in feet:
1 answer:
The complete question in the attached figure
the correct vertex is (50,75)
we have that
<span>x intercepts at ordered pairs (0, 0) and (100, 0)
The vertex is at (50, 75)
we know that
the general equation of a parabola is
y=ax</span>²+bx+c
<span>
for x=0--------> y=0
so
c=0
for x=100-------> y=0
so
0=a*100</span>²+b*100-------> 0=a*10000+b*100-------> 0=100*a+b
b=-100*a-------> equation 1
for x=50---------> y=75
so
75=a*50²+b*50--------> 75=a*2500+b*50-------> 1=33.33*a+0.67*b
1=33.33*a+0.67*b----------> equation 2
using a graph tool----------> to resolve the system
the solution is
a=-0.3
b=2.97
the equation is
y=-0.03x²+2.97x
Part A:
1) What do the x-intercepts and maximum value of the graph represent?
The x-intercepts represent the points at the ground, so with that you have the width of the tunnel at ground level is 100 ft - 0 ft = 100 ft.
The vertex represents the maximum height, which is at the center of the tunnel. So, you know that the maximum height is at x = 50 ft and it is 75 ft
2) What are the intervals where the function is increasing and decreasing, and what do they represent about the distance and height?
The increasing interval is (0,50), and the decreasing interval is (50,100).
That represents that the height increase from the left border (x = 0) until the center of the tunnel (x = 50 ft), reaching a height of 75 ft, and starts to decrease toward the right border until the ground level (x = 100).
<span>Part B:
3) What is an approximate average rate of change of the graph from x = 15 to x = 35, and what does this rate represent?
</span>Find the y -coordinate for x = 15 and for x = 35.
for x=15
y=y=-0.03x²+2.97x------> y=37.8
for x=35
y=y=-0.03x²+2.97x------> y=67.2
Then the average rate of change is equal to [change in height] / [change in x] = [67.2-37.8] / [35 - 15] = 29.4 / 20 = 1.47
1.47 > 0
the number is positive means that the height is increasing in that interval
the correct vertex is (50,75)
we have that
<span>x intercepts at ordered pairs (0, 0) and (100, 0)
The vertex is at (50, 75)
we know that
the general equation of a parabola is
y=ax</span>²+bx+c
<span>
for x=0--------> y=0
so
c=0
for x=100-------> y=0
so
0=a*100</span>²+b*100-------> 0=a*10000+b*100-------> 0=100*a+b
b=-100*a-------> equation 1
for x=50---------> y=75
so
75=a*50²+b*50--------> 75=a*2500+b*50-------> 1=33.33*a+0.67*b
1=33.33*a+0.67*b----------> equation 2
using a graph tool----------> to resolve the system
the solution is
a=-0.3
b=2.97
the equation is
y=-0.03x²+2.97x
Part A:
1) What do the x-intercepts and maximum value of the graph represent?
The x-intercepts represent the points at the ground, so with that you have the width of the tunnel at ground level is 100 ft - 0 ft = 100 ft.
The vertex represents the maximum height, which is at the center of the tunnel. So, you know that the maximum height is at x = 50 ft and it is 75 ft
2) What are the intervals where the function is increasing and decreasing, and what do they represent about the distance and height?
The increasing interval is (0,50), and the decreasing interval is (50,100).
That represents that the height increase from the left border (x = 0) until the center of the tunnel (x = 50 ft), reaching a height of 75 ft, and starts to decrease toward the right border until the ground level (x = 100).
<span>Part B:
3) What is an approximate average rate of change of the graph from x = 15 to x = 35, and what does this rate represent?
</span>Find the y -coordinate for x = 15 and for x = 35.
for x=15
y=y=-0.03x²+2.97x------> y=37.8
for x=35
y=y=-0.03x²+2.97x------> y=67.2
Then the average rate of change is equal to [change in height] / [change in x] = [67.2-37.8] / [35 - 15] = 29.4 / 20 = 1.47
1.47 > 0
the number is positive means that the height is increasing in that interval
You might be interested in
You have 2 apples, get 3 more, and then lose 2 of them.
So, you HAVE 2 apples, and if you get 3 more, you can add it to the 2.
In total at the moment, you have 5 apples. Now, you lose 2 more. We can subtract that from our total number of apples.
We now know that you are left with 3 apples.
So, you HAVE 2 apples, and if you get 3 more, you can add it to the 2.
In total at the moment, you have 5 apples. Now, you lose 2 more. We can subtract that from our total number of apples.
We now know that you are left with 3 apples.
At the start, the tank contains
(0.02 g/L) * (1000 L) = 20 g
of chlorine. Let <em>c</em> (<em>t</em> ) denote the amount of chlorine (in grams) in the tank at time <em>t </em>.
Pure water is pumped into the tank, so no chlorine is flowing into it, but is flowing out at a rate of
(<em>c</em> (<em>t</em> )/(1000 + (10 - 25)<em>t</em> ) g/L) * (25 L/s) = 5<em>c</em> (<em>t</em> ) /(200 - 3<em>t</em> ) g/s
In case it's unclear why this is the case:
The amount of liquid in the tank at the start is 1000 L. If water is pumped in at a rate of 10 L/s, then after <em>t</em> s there will be (1000 + 10<em>t</em> ) L of liquid in the tank. But we're also removing 25 L from the tank per second, so there is a net "gain" of 10 - 25 = -15 L of liquid each second. So the volume of liquid in the tank at time <em>t</em> is (1000 - 15<em>t </em>) L. Then the concentration of chlorine per unit volume is <em>c</em> (<em>t</em> ) divided by this volume.
So the amount of chlorine in the tank changes according to
which is a linear equation. Move the non-derivative term to the left, then multiply both sides by the integrating factor 1/(200 - 5<em>t</em> )^(5/3), then integrate both sides to solve for <em>c</em> (<em>t</em> ):
There are 20 g of chlorine at the start, so <em>c</em> (0) = 20. Use this to solve for <em>C</em> :
