Aballoon is rising vertically above a level, straight road at a constant rate of 1 ft/sec. just when the balloon is 65 feet above the ground, a bicycle passes under it, going 17 ft/sec. how fast is the distance between and bicycle and the balloon changing 3 seconds later?

DL/dt  =  11 feet/sec

Step-by-step explanation:

We have a right triangle between    balloon   ( h  =  65 feet)  from the point straight down the ballon,  and the bicycle position. Hypothenuse is L the distance between balloon and bicycle

Let  x be distance between (vertical from the balloon to ground) then

according to Pythagoras Theorem

L²  =  x² +  h²       (1)

and we know

Dx/dt  =  17   ft/sec

Dh/dt =  1  ft/sec

We need to find   DL/dt

From equation (1)

L²   =  x²   +  h²

Differentiating on both sides of the equation

2L DL/dt   =  2xDx/dt  +  2yDy/dt

L*DL/dt   =  x*Dx/dt + y*Dy/dt

DL/dt    =   [  x*Dx/dt + y*Dy/dt ] /L    (2)

Now  3 sec after the bicycle passed under the balloon

x  =  3 * Dx/dt     =  3 * 17   = 51 feet

and  h  =  65 + 1* 3    =  68 feet

then   L  = √ √(68)²  + (51)²         L  =  √ 4624  +  2601    L  = 85 feet

Now plugging in these values in equation  (2)

DL/dt  =  [ 51*17  +  68* 1]  / 85

DL/dt  =  11 feet/sec

the range is 19

step-by-step explanation:

the range is biggest number minus the smallest, so in this case it would be

41 - 22 = 19

step-by-step explanation:

we are given a graph for system of inequalities.

in order to write the system of inequalities for the given graph, we need to find the inequality for each of the line separately.

let us find the slope of dotted line first.

taking two coordinates on the graph.

let us say (0,0) and (3,1).

let us find the slope of the line now.

slope of the dotted line would be =.

y-intercept of dotted line is y=0 only.

plugging slope m= and b=0 in slope-intercept form y=mx+b, we get

.

the line is shaded up of that dotted line.

therefore, for shading up we would take greater than inequality symbol > .

and because it's a dotted line we would take only > symbol in place of = sign in above equation of line.

therefore, inequality for dotted line would be

let us take two points (3,1) and (0,4) on the solid line.

slope of the solid line would be =.

for the solid line we have y-intercept b= 4.

plugging slope m= -1 and b=4 in slope-intercept form y=mx+b, we get

y = -1x + 4.

the line is shaded down of that solid line.

therefore, for shading down we would take less than inequality symbol < .

and because it's a solid line we would take ≤ symbol in place of = sign in above equation of line.

therefore, inequality for solid line would be