A right triangle has legs, so the rate of change of the area is = -102.5 [tex]\frac{inches^{2} }{sec}[/tex].
A right triangle has legs of 15 inches and 20 inches
The short leg is decreasing by 4 in/sec and the long leg is shrinking at 7 in/sec.
Let us assume,
Two legs are a and b
So ,
We can write,
a = 15 - 5t
b = 20 - 7t
The area of a right triangle is the region occupied inside the boundary of the right-angled triangle.
The formula for the area of a right-angle triangle is A = (½)× b× h square units.
If you know the two legs, then use the formula area = a × b / 2, where a, and b are the legs.
The rate of change of the area,
area A = ab /2
A = [tex]\frac{(15 - 5t)*(20 - 7t)}{2}[/tex]
We can multiply the coefficients,
A = 300 - 105t - 100t + 35[tex]t^{2}[/tex] /2
A = 35[tex]t^{2}[/tex] - 205t +300 / 2
A = 35/2[tex]t^{2}[/tex] - 205/2t + 300/2
A = 17.5[tex]t^{2}[/tex] - 102.5t + 150
The rate of change of the area is,
[tex]\frac{dA}{dt}[/tex] = d/dt ( 17.5[tex]t^{2}[/tex] - 102.5t + 150)
[tex]\frac{dA}{dt}[/tex] = 35t - 102.5
And at t=0,
[tex]\frac{dA}{dt}[/tex] = 35(0) - 102.5
[tex]\frac{dA}{dt}[/tex] = -102.5 [tex]\frac{inches^{2} }{sec}[/tex]
Therefore,
A right triangle has legs, so the rate of change of the area is = -102.5 [tex]\frac{inches^{2} }{sec}[/tex].
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