Physics, 31.07.2019 12:30 smagallanes

A15-foot ladder is leaning against a vertical wall. if the bottom of the ladder is being pulled away from the wall at the rate of 7 feet per second, at what rate is the area of the triangle formed by the wall, the ground, and the ladder changing, in square feet per second, at the instant the bottom of the ladder is 9 feet from the wall?

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This is actually a calculus question. so, basically, this is a related rates question where you are given 1 quantity and you have to find the other. we'll define some variables first.x = 9 ftdx/dt = 7 ft/secl = 15 ftsince you want to find the area, you have to find a formula for it. which is: $a = \frac{1}{2}x*h$ we know the base (x), but we don't know the height (h). we can figure it out by solving for it. we have to form an equation which we can figure out height from given the information we have. this equation is: $x^2+h^2=l^2$ plug in known values: $9^2+h^2=15^2$ solve for h: $h^2 = 15^2-9^2$ $h^2=144$ $h = \sqrt{144}=12$ now, let's solve for the rate at which h is changing. we can do so by differentiating the pythagorean theorem (above). let's do that: $x^2+h^2=15$ $2x \frac{dx}{dt}+2h \frac{dh}{dt}=0$ $\frac{dh}{dt} = \frac{-2x \frac{dx}{dt} }{2h}$ $\frac{dh}{dt} = \frac{-x \frac{dx}{dt} }{h}$ plug in values: $\frac{dh}{dt} = \frac{-9*7}{12} = \frac{-63}{12} = \frac{-21}{4}$ now we have all the given information to do this problem. so, to figure out the rate of change of area, we can differentiate (with respect to time) the earlier equation: $\frac{d}{dt}(a) = \frac{d}{dt} (\frac{1}{2}x*h)$ which is equal to: $\frac{da}{dt}= \frac{1}{2}( \frac{dx}{dt}h+x \frac{dh}{dt})$ plug in known values: $\frac{da}{dt} = \frac{1}{2}(7*12+9* \frac{-21}{4} ) = \frac{1}{2}(84- \frac{189}{4} )$ $\frac{1}{2}( \frac{147}{4}) = \frac{147}{8}$ so, the area of the triangle is changing at approximately 18.375ft²/sechope this all makes sense. if it doesn't, either contact me, or watch khan academy - he explains it real well. special credit to turtaniumx and quietself for letting me know of the error, and fix it!

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Neptune is the eighth planet from the sun. it was the first planet to get its existence predicted by mathematical calculations before it was actually seen through a telescope on sept. 23, 1846. irregularities in the orbit of uranus led french astronomer alexis bouvard to suggest that the gravitational pull from another celestial body might be responsible. german astronomer johann galle then relied on subsequent calculations to spot neptune via telescope. previously, astronomer galileo galilei sketched the planet, but he mistook it for a star due to its slow motion. in accordance with all the other planets seen in the sky, this new world was given a name from greek and roman mythology — neptune, the roman god of the sea.

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Ireally don’t know

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A15-foot ladder is leaning against a vertical wall. if the bottom of the ladder is being pulled away...

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