Question

he Hobbits are building a watchtower so they can prepare to battle in case trolls decide to attack them. One Hobbit will always be on the lookout and the Hobbits will prepare for battle as soon as the trolls are visible. The time, TTT, in minutes, that the Hobbits have to prepare for the attack of trolls kkk meters away is given by the function T(k)=\dfrac{k}{80}T(k)= 80 k ​ T, left parenthesis, k, right parenthesis, equals, start fraction, k, divided by, 80, end fraction. The visibility, VVV, in meters, that the Hobbits have from an mmm meter watch tower is given by the function V(m)=50mV(m)=50mV, left parenthesis, m, right parenthesis, equals, 50, m. Find an explicit expression that models the amount of time the Hobbits have to prepare for a troll attack given that the watchtower is mmm meters tall.

Answer 1

5m/8

Step-by-step explanation:

Function T gives the time the Hobbits have to prepare for the attack, T(k), in minutes, as a function of troll's distance, k, in meters.

Function V gives visibility from the watchtower, V(m), in meters, as a function of the height of the watchtower, m, in meters.

Therefore, T(V(m)) will give the time the Hobbits have to prepare for the troll attack as a function of the height, m, of the watchtower.

We can input m into function V to obtain the visibility from watchtower, V(m), in meters. Since visibility indicates the distance you can see, this also gives the distance of the trolls. This can then be input into function T to obtain the time that the Hobbits have to prepare for a troll attack.

Let's find T(V(m)) by substituting the formula for V(m) into function T as shown below.

T(V(M))=T(50m)

=50m/80

We can simplify this as follows:

=50m/80

=5m/8