The maximum velocity of a roller coaster depends on the vertical drop from the top of the highest hill to the bottom of that hill. The formula:V(h)=sqrt(h-2) , gives the relationship between maximum velocity, V(h) in feet per second, and height, h in feet.
1. Identity the independent variable, dependent variable.
2. What is the domain of the radical function V?
3. What is the maximum velocity V(h) of the roller coaster when the height h is 66 feet? (Specify and explain your answer in words.
4. What is the height h of the roller coaster, if its maximum velocity V(h) is 32 feet per second? Specify and explain your answer in words.
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1. The independent variable in this case is the height of the roller coaster (h), while the dependent variable is the maximum velocity (V).
2. The domain of the radical function V is determined by the requirement that the height (h) must be greater than or equal to 2, because otherwise the radical expression would produce a negative number, which is not defined in the real number system.
3. To find the maximum velocity V(h) when h is 66 feet, we can substitute this value into the formula: V(66) = sqrt(66-2). Evaluating this expression gives us V(66) = sqrt(64) = 8 feet per second. Therefore, when the height of the roller coaster is 66 feet, the maximum velocity is 8 feet per second.
4. To find the height (h) of the roller coaster when the maximum velocity V(h) is 32 feet per second, we can rearrange the formula and solve for h. Rewriting the formula as h = V^2 + 2, we can substitute V = 32 into this equation: h = 32^2 + 2. Simplifying, we get h = 1026 feet. Therefore, when the maximum velocity of the roller coaster is 32 feet per second, the height of the roller coaster is 1026 feet.