Answer:
a. f(t) = h = 10·t/(π·r²)
b. The dependent and independent variables are ;
Dependent = The height of the water in the cup over time, h
Independent = The time of measurement of the water height in the cup
c. The domain is 0 < t < 10
d. The 5 in f(5), represents 5 seconds
e. Susan was trying to find out what the height of the water in the cup will be after 5 seconds
f. The numbers mean that at 7 seconds the water in the cup is 70 units high
Step-by-step explanation:
a. The rate at which water is added to the cup in the question = 10 mL per second
The volume of water in the cup is V = π × r² × h
dV/dt = 10
dV = 10 × dt
V = ∫10 × dt = 10·t
The height of the water in the cup, h = V/(π × r²) = 10·t/(π·r²)
Therefore, we have the function that models the height of the water in the cup over time given as follows;
f(t) = h = 10·t/(π·r²)
b. The dependent and independent variables are given as follows;
Dependent variable = The height of the water in the cup over time, h
Independent variable = The time of measurement of the water height in the cup
c. The domain is 0 < t < 10
Given that the cup has a capacity of 100 mL, we have;
The maximum volume the up can hold = 100 mL
The time it will take to fill the 100 mL cup given that the rate of at which she is pouring water into the cup at 10 mL per second = 100 mL/(10 mL/s) = 10 s
Therefore, a reasonable domain for the independent variable, t is 0 < t < 10
d. We have that the function f(t) = h = 10·t/(π·r²)
Therefore, the 5 in f(5), represents the time in seconds at which the function was being evaluated, which is 5 seconds
e. Given that Susan wrote f(5), Susan was trying to find the height of the water in the cup after 5 seconds
f. Given that Susan wrote f(7) = 70, we have that at 7 seconds the height of the water in the cup is 70 units, where one unit is 1/(π·r²)
Which gives;
f(7) = h = 10 × 7/(π·r²) 70/(π·r²)
f(7) = 70/(π·r²)
