Answer to Question #209555 in Calculus for Delmundo

Question #209555

1. Determine the equation of the family of the curves whose slope is

3x – 5.

Also find the equation of the member whose curve passes through point

(1, 1).

2. A body is thrown vertically upward from the ground with an initial velocity

of 64 ft/sec. Find the maximum height attained by the body. Regardless of

frictional force, determine the total time the body returns to the ground.

(note: a = 32ft/sec2)


1
Expert's answer
2021-06-23T04:48:58-0400

1. The slope of the curve is given by 


slope=m=dydx=3x5slope=m=\dfrac{dy}{dx}=3x-5

dy=(3x5)dxdy=(3x-5)dx

Integrate both sides with respect to xx 


dy=(3x5)dx\int dy=\int(3x-5) dxy=32x25x+Cy=\dfrac{3}{2}x^2-5x+C


The equation of the family of the curves is


y=32x25x+Cy=\dfrac{3}{2}x^2-5x+C

As the curve passes through the point (1,1)(1,1) we have,


1=32(1)25(1)+C1=\dfrac{3}{2}(1)^2-5(1)+CC=92C=\dfrac{9}{2}

The equation of the curve pasiing through the point (1,1)(1,1) is


y=32x25x+92y=\dfrac{3}{2}x^2-5x+\dfrac{9}{2}

1.


v(t)=v0atv(t)=v_0-at

The equation of the motion of the body along the positive y-axis is


y(t)=y0+v0tat22y(t)=y_0+v_0t-\dfrac{at^2}{2}

The body attains the maximum height when v=0v=0


v0at=0=>t=v0av_0-at=0=>t=\dfrac{v_0}{a}

Substitute


ymax=y(v0a)=y0+v02av022a=y0+v022ay_{max}=y(\dfrac{v_0}{a})=y_0+\dfrac{v_0^2}{a}-\dfrac{v_0^2 }{2a}=y_0+\dfrac{v_0^2}{2a}

Given y0=0,v0=64 ft/sec,a=32 ft/sec2y_0=0, v_0=64\ ft/sec, a=32\ ft/sec^2

ymax=0 m+(64 ft/sec)22(32 ft/sec2)=64 fty_{max}=0 \ m+\dfrac{(64\ ft/sec)^2}{2(32\ ft/sec^2)}=64\ ft

The body returns to the ground when y(t)=0,t>0y(t)=0, t>0


v0tat22=0,t>0v_0t-\dfrac{at^2}{2}=0, t>0

v0at2=0v_0-\dfrac{at}{2}=0

t=2v0at=\dfrac{2v_0}{a}

t=2(64 ft/sec)32 ft/sec2=4 sect=\dfrac{2(64\ ft/sec)}{32\ ft/sec^2}=4\ sec

The maximum height attained by the body is 64 ft.64\ ft.

The body returns to the ground 4 sec4\ sec after the start.



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