A trough is 10 ft long and its ends have the shape of isosceles triangles that are 3 ft across at the top and have a height of 1 ft. If the trough is being filled with water at a rate of 12ft3min, how fast is the water level rising when the water is 6 inches deep?

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Natalie Yamamoto · Accepted Answer

Step 1  When the water is x feet deep, let us assume that the isosceles cross section formed by the water is b feet wide  Using the property of similar triangles, we can write  b3=x1  Multiply both sides by 3  b=3x  The area of the cross section formed by the water is  A=12(base)(height)  Substitute 3x in place for base and x for height  A=3x22  Volume of the water=(length of the trough)×(Area of the cross section)  V(x)=10×3x22=15x2  Step 2  To find the rate at which the water level is rising, differentiate with respect to time  dVdt=d(15x2)dt  Use the chain rule in the right-hand side  dVdt=d(15x2)dx×dxdt  Use the power rule to differentiate 15x2  dVdt=15×2x2−1×dxdt=30x×dxdt  Substitute x=0.5 because 6inches=0.5feet and dVdt=12  12=30×0.5×dxdt  12=15×dxdt  dxdt=1215=45=0.8ftmin  The water level is rising at the rate of 0.8ftmin

veiga34 · Accepted Answer

Step 1 Given Information: V=b×h×l2 dVdt=12ft3min when h=6 inchs⇒h=0.5ft dhdt=? 31=bh⇒b=3h v=b×h×102⇒v3h×h×102⇒V=15h2 Step 2 dVdt=dVdh×dhdt Sub in for h=0.5ft and dVdt=12ft3min dhdt=45=0.8ftmin

nick1337 · Accepted Answer

Step 1Let&amp;nbsp;V&amp;nbsp;(t)&amp;nbsp;=he volume of water in the trough at time t.We knowdVdt=12ft3minsince the amount of water in the trough is increasing (“filled”) at the rate of12ft3minLet&amp;nbsp;h=the “depth” of water in the trough at time t. Note that this depth is in fact the height of the triangle that the water makes on the end of the trough.We want&amp;nbsp;dhdtwhen&amp;nbsp;h=12ftWe need an equation relating V and h.V=(area of base of trough)×(height of trough)=(area of the∆of water)×(10 ft)=(12×base of the∆of water×height of the∆of water)×(10&amp;nbsp;ft)To find the base of the&amp;nbsp;∆of water in terms of h, we use the similar triangles shown at right.=(12×3&amp;nbsp;h×h)(10)=15h2We differentiate with respect to tddt[V]=ddt[15h2]dVdt=30h×dhdtsubstituting12=30×12dhdt1215=dhdtor45=dhdtThus the water level is rising at the rate of&amp;nbsp;45ftminwhen the water is 6 inches deep.

A trough is 10 ft long and its ends have the shape of isosceles triang

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