# The proportionality equation through given conditions.

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 1, Problem 22T

(a)

To determine

## The proportionality equation through given conditions.

Expert Solution

The equation of proportionality is M=kwh2/L .

### Explanation of Solution

Given:

The maximum weight M is proportional to the width w of beam and square height h and inversely proportional of length of beam L.

Calculation:

From given conditions, the maximum weight M that supported by a beam is directly proportional to width w of beam and square of height so by the definition of direct proportionality the equation is,

M=k1wh2 . (1)

For some constant k10 .

The weight w is inversely proportional to length of beam L so by the definition of inverse proportionality the equation is,

M=k2L . (2)

For some constant k20

Combine both equation (1) and (2) to get the proportionality equation,

M=k1k2wh2L=kwh2L

For some constant k0

Where,

k is constant of proportionality.

In the above equation k1 and k2 are constant so the product of constant is a constant.

Thus, the equation of proportionality is M=kwh2/L .

(b)

To determine

### The value of constant of proportionality through given conditions.

Expert Solution

The value of constant of proportionality is 400.

### Explanation of Solution

Given:

The width of beam is 4in , height of beam is 6in and length of beam is 12ft that support a weight of 4800Ib .

Calculation:

The equation of proportionality from part (a) is,

M=kwh2/L

Substitute 4 for w, 6 for h, 12 for l and 4800 for M in the above equation and solve for k,

4800=k(4)(6)212k(4)(6)2=480012k=57600144k=400

Thus, the value of constant of proportionality is 400.

(c)

To determine

### The maximum weight beam can support.

Expert Solution

The value of maximum weight is 12,000Ib .

### Explanation of Solution

Given:

The length of beam is 10-ft , width of beam is 3in and height of beam is 10in because the beam is made of same material that means the value of constant of proportionality is k=400 same for this beam obtained from part (b).

Calculation:

The equation of proportionality from part (a) is,

M=kwh2/L

Substitute 3 for w, 10 for h, 10 for l and 400 for k in the above equation and solve for M,

M=(400)(3)(10)2/10M=12,0000/10M=12,000

Thus, the value of maximum weight is 12,000Ib .

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