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Helium is pumped into a spherical balloon at a constant rate of 2 cubic feet per second. How fast is the radius increasing after 3 minutes?. At what time (if any) is the radius increasing at a rate of 120 feet per second? (Volume of a sphere: \( V=\frac{4}{3} \pi r^{3} \) ) The radius is increasing at a rate of \( 0.0082 \mathrm{ft} / \mathrm{sec} \). (Type an integer or a decimal. Do not round until the final answer. Then round to four decimal places as needed.) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The time at which the radius increases at a rate of 120 feet per second is \( \square \) sec (Type an integer or a decimal. Do not round until the final answer. Then round to five decimal places as needed.) B. There is no solution.

Ask by Wade Todd. in the United States
Mar 18,2025

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The radius is increasing at a rate of 0.0082 feet per second after 3 minutes. A. The time at which the radius increases at a rate of 120 feet per second is 0.000098 seconds.

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Extra Insights

Helium balloons have a fascinating history! They first flew high into the skies in the late 18th century. The first manned flight in a hot air balloon took place in 1783, piloted by Jean-François Pilâtre de Rozier. Fast forward to today, and helium balloons are used not just for parties but also for scientific research and even weather monitoring! In the real world, balloons can expand dramatically, but they also have limits. If you keep pumping in helium, pressure increases, which can lead to popping! It's essential to know the geometry behind them to determine how big they can get without facing an explosive end. So, if you're feeling scientific, remember to use the right formulas to understand these buoyant beauties!

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