The radius of a sphere-shaped balloon increases at a rate of 2 centimeters $(cm)$ per second. If the surface area of the completely inflated balloon is $784 \pi cm^2$, how long will it take for the balloon to fully inflate?
Use $S A=4 \pi r^2$.
A. 7 seconds
B. 10 seconds
C. 49 seconds
D. 196 seconds
Answer (2)
Find the radius of the fully inflated balloon using the surface area formula: 4 π r 2 = 784 π , which gives r = 14 cm.
Express the radius as a function of time, given the rate of increase: r ( t ) = 2 t .
Set the radius function equal to the final radius and solve for time: 2 t = 14 .
Calculate the time it takes to fully inflate the balloon: t = 7 seconds. The answer is 7 .
Explanation
Problem Analysis We are given that the radius of a sphere-shaped balloon increases at a rate of 2 cm/s. The surface area of the fully inflated balloon is 784 π c m 2 . We need to find how long it will take for the balloon to fully inflate. The formula for the surface area of a sphere is S A = 4 π r 2 .
Finding the Radius First, we need to find the radius of the fully inflated balloon. We are given that the surface area is 784 π c m 2 . So we have:
4 π r 2 = 784 π
Simplifying the Equation Divide both sides of the equation by 4 π :
r 2 = 4 π 784 π = 196
Calculating the Radius Take the square root of both sides to find the radius:
r = 196 = 14 cm
Radius as a Function of Time Since the radius increases at a rate of 2 cm/s, we can write the radius as a function of time t as:
r ( t ) = 2 t
Setting up the Time Equation We want to find the time t when the radius is 14 cm. So, we set r ( t ) = 14 :
2 t = 14
Solving for Time Divide both sides by 2 to solve for t :
t = 2 14 = 7 seconds
Final Answer Therefore, it will take 7 seconds for the balloon to fully inflate.
Examples
Imagine you're inflating a bouncy ball for a game. Knowing how quickly the radius increases helps you predict when it will reach the right size for optimal bouncing. This is similar to understanding rates of change in calculus, where we relate how fast one quantity (like the radius) changes with respect to another (like time). By understanding these relationships, you can accurately predict outcomes in various real-world scenarios, from inflating sports equipment to understanding growth rates in biological systems.
It will take 7 seconds for the balloon to fully inflate. We found this by determining the radius from the surface area and then calculating the time it takes for the radius to reach that size, given the rate of increase. Thus, the answer is 7 seconds.
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