# Calculus Question Of The Day

**Calculus 12 question how wide must the gondola be?**

The designer of a 30 m diameter, hot air balloon wishes to suspend the gondola 8 m below the bottom of the balloon with suspension cables tangent to the surface of the balloon. Two of the cables are shown running from the top edges of the gondola to their points of tangency (-12, -9) and (12,-9). How wide must the gondola be?

I’ve included a picture of the daigram.

http://s1008.photobucket.com/albums/af208/jotti_86gill/?action=view¤t=scan0001.jpg

Please help me I have been trying to solve this question for days but am getting nowhere.

i have to use derivatives to answer this question

To make my solution understandable, I’ll describe the sketch I made.

Label The center of the balloon as O.

Draw a vertical line from O downwards.

Let T be one of the point of tangency, and draw TB, a tangent line intersecting the vertical line at B.

Draw a horizontal line from T to OB and label the intersection as S.

Label one end of the gondola as W and the horizontal line from W to the line OB.

The intersection of the gondola line with OB as U.

ST = 12

OT = 9

OT = 15 (radius)

By similar triangles:

OB : OT : : OT : OS

OB : 15 : : 15 : 9

OB = 15 × 15 / 9 = 25

UB = OB – (15 + 8) = 25 – 23 = 2

SB = 25 – 9 = 16

UW : UB : : ST : SB

UW : 2 : : 12 : 16

UW = 2 × 12 /16 = 24/16 = 3/2

w = 2(UW) = 2(3/2) = 3 meters

**Mathematics – Multivariable Calculus – Lecture 2**