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Recall
Connected rates of change — objective 1
Maths4U Original · Short practice
A spherical bubble has radius \(\displaystyle r=3+2t \mathrm{cm}\), where t is measured in seconds. Let \(\displaystyle A=\pi r^{2}\) be its circular cross-sectional area and \(\displaystyle V= \frac{4\pi r^{3}}{3}\) its volume. Evaluate all instantaneous rates at radius 8 cm.
Find \(\displaystyle \frac{\frac{dA}{dt}}{\pi}\) when the radius is 8 cm.
Enter one exact number or fraction. These checks support non-calculator practice. Give full reasoning separately for written problems.
Fresh review variant
Connected rates of change — objective 1
Maths4U Original · Short practice
A spherical bubble has radius \(\displaystyle r=1+3t \mathrm{cm}\), where t is measured in seconds. Let \(\displaystyle A=\pi r^{2}\) be its circular cross-sectional area and \(\displaystyle V= \frac{4\pi r^{3}}{3}\) its volume. Evaluate all instantaneous rates at radius 6 cm.
Find \(\displaystyle \frac{\frac{dA}{dt}}{\pi}\) when the radius is 6 cm.
Enter one exact number or fraction. These checks support non-calculator practice. Give full reasoning separately for written problems.
Method
Connected rates of change — objective 2
Maths4U Original · Short practice
A spherical bubble has radius \(\displaystyle r=1+3t \mathrm{cm}\), where t is measured in seconds. Let \(\displaystyle A=\pi r^{2}\) be its circular cross-sectional area and \(\displaystyle V= \frac{4\pi r^{3}}{3}\) its volume. Evaluate all instantaneous rates at radius 6 cm.
Find \(\displaystyle \frac{\frac{dV}{dt}}{\pi}\) at the same radius.
Enter one exact number or fraction. These checks support non-calculator practice. Give full reasoning separately for written problems.
Fresh review variant
Connected rates of change — objective 2
Maths4U Original · Short practice
A spherical bubble has radius \(\displaystyle r=2+4t \mathrm{cm}\), where t is measured in seconds. Let \(\displaystyle A=\pi r^{2}\) be its circular cross-sectional area and \(\displaystyle V= \frac{4\pi r^{3}}{3}\) its volume. Evaluate all instantaneous rates at radius 7 cm.
Find \(\displaystyle \frac{\frac{dV}{dt}}{\pi}\) at the same radius.
Enter one exact number or fraction. These checks support non-calculator practice. Give full reasoning separately for written problems.
Apply
Connected rates of change — objective 3
Maths4U Original · Short practice
A spherical bubble has radius \(\displaystyle r=2+4t \mathrm{cm}\), where t is measured in seconds. Let \(\displaystyle A=\pi r^{2}\) be its circular cross-sectional area and \(\displaystyle V= \frac{4\pi r^{3}}{3}\) its volume. Evaluate all instantaneous rates at radius 7 cm.
Find the time in seconds to reach this radius.
Enter one exact number or fraction. These checks support non-calculator practice. Give full reasoning separately for written problems.
Fresh review variant
Connected rates of change — objective 3
Maths4U Original · Short practice
A spherical bubble has radius \(\displaystyle r=3+1t \mathrm{cm}\), where t is measured in seconds. Let \(\displaystyle A=\pi r^{2}\) be its circular cross-sectional area and \(\displaystyle V= \frac{4\pi r^{3}}{3}\) its volume. Evaluate all instantaneous rates at radius 8 cm.
Find the time in seconds to reach this radius.
Enter one exact number or fraction. These checks support non-calculator practice. Give full reasoning separately for written problems.
Check understanding
Connected rates of change — objective 4
Maths4U Original · Short practice
A spherical bubble has radius \(\displaystyle r=3+1t \mathrm{cm}\), where t is measured in seconds. Let \(\displaystyle A=\pi r^{2}\) be its circular cross-sectional area and \(\displaystyle V= \frac{4\pi r^{3}}{3}\) its volume. Evaluate all instantaneous rates at radius 8 cm.
Find \(\displaystyle \frac{dV}{d}A\) at this radius.
Enter one exact number or fraction. These checks support non-calculator practice. Give full reasoning separately for written problems.
Fresh review variant
Connected rates of change — objective 4
Maths4U Original · Short practice
A spherical bubble has radius \(\displaystyle r=1+2t \mathrm{cm}\), where t is measured in seconds. Let \(\displaystyle A=\pi r^{2}\) be its circular cross-sectional area and \(\displaystyle V= \frac{4\pi r^{3}}{3}\) its volume. Evaluate all instantaneous rates at radius 6 cm.
Find \(\displaystyle \frac{dV}{d}A\) at this radius.
Enter one exact number or fraction. These checks support non-calculator practice. Give full reasoning separately for written problems.