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Chapter 2: Q. 40 (page 125)

Constructing a Closed Box A closed box with a square base is required to have a volume of 10cubic feet.

(a) Build a model that expresses the amount Aof material used to make such a box as a function of the length xof a side of the square base.

(b) How much material is required for a base 1foot by 1foot?

(c) How much material is required for a base 2feet by 2feet?

(d) Graph A=A(x). For what value of xis Asmallest?

Short Answer

Expert verified

(a) The amount Aof material used to make such a box is A(x)=2x2+40x

(b) Material required for a base 1foot by

1foot is 42 foot2.

(c) Material required for a base 2foot by

2foot is 28 foot2.

(d) The value ofAis smallest at103feet.

Step by step solution

01

Step 1. Part (a) Build a model that expresses the amount A of material used to make such a box as a function of the length x of a side of the square base.

Let hbe the height of the box and xbe the side of a square base.

Volume=base area脳height

V=x2hh=Vx2

Since V=10, we get

h=10x2

Now the amount Aof material used to make such a box is

A=2脳areaofthebase+4脳areaofthesideA=2x2+4hx

Since h=10x2,

A(x)=2x2+410x2xA(x)=2x2+40x

02

Step 2. Part (b) Find material required for the box of base 1 foot by 1 foot.

A(1)=2(1)2+401A(1)=2+40A(1)=42foot2

03

Step 3. Part (c) Find material required for the box of base 2 foot by 2 foot.

A(2)=2(2)2+402A(2)=8+20A(2)=28foot2

04

Step 4. Part (d) Draw the graph of A(x) and find the value of x where A is smallest.

The graph is

The value of xwhere A is smallest isx=2.15feet(accurate value is103feet)

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W={t0鈮�v<1.7933-(10.45+10v-v)(33-t)22.04,1.79鈮�v鈮�2033-1.5958(33-t),v>20

where vrepresents the wind speed (in meters per second) and trepresents the air temperature . Compute the wind chill for the following:

(a) An air temperature of localid="1645962228946" 10掳Cand a wind speed of 1 meter per second localid="1645962235368" (m/sec)

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(c) An air temperature of localid="1645962254138" 10掳Cand a wind speed of localid="1645962259043" 15m/sec

(d) An air temperature of localid="1645962265610" 10掳Cand a wind speed of localid="1645962274629" 25m/sec

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