The mass of a sheet of metal varies jointly with its area and its thickness. A sheet of metal of area [tex]$250 cm^2$[/tex] and thickness 1 mm has a mass of 200 g. Find the formula which connects the mass M g, the area [tex]$Acm ^2$[/tex], and the thickness t mm. Hence find the mass of a piece of metal of area [tex]$400 cm^2$[/tex] and thickness 3 mm.
Answer (1)
Establish the relationship between mass M , area A , and thickness t as M = k A t .
Use the given values M = 200 g , A = 250 c m 2 , and t = 1 mm to find the constant of proportionality k = 0.8 .
Substitute the value of k back into the formula to get M = 0.8 A t .
Use the formula to find the mass M when A = 400 c m 2 and t = 3 mm , resulting in M = 960 g .
960 g
Explanation
Understanding the Problem We are given that the mass M of a sheet of metal varies jointly with its area A and its thickness t . This means that there exists a constant k such that M = k A t . We are given that a sheet of metal with area A = 250 c m 2 and thickness t = 1 mm has a mass of M = 200 g . We need to find the formula connecting M , A , and t , and then find the mass M of a piece of metal with area A = 400 c m 2 and thickness t = 3 mm .
Setting up the Equation Since M varies jointly with A and t , we can write the relationship as M = k A t , where k is the constant of proportionality.
Finding the Constant of Proportionality We are given M = 200 g , A = 250 c m 2 , and t = 1 mm . We can use these values to find the constant of proportionality k . Substituting the values into the equation, we get:
200 = k "."250"."1
Calculating k Solving for k , we have:
k = 250"."1 200 = 250 200 = 5 4 = 0.8
The Formula Now that we have the value of k , we can write the specific formula for this metal as:
M = 0.8 A t
Finding the Mass We are asked to find the mass M when A = 400 c m 2 and t = 3 mm . Substituting these values into the formula, we get:
M = 0.8"."400"."3
Calculating the Final Mass Calculating the mass, we have:
M = 0.8"."400"."3 = 0.8"."1200 = 960
So, the mass of the piece of metal is 960 g.
Final Answer Therefore, the formula connecting the mass M , the area A , and the thickness t is M = 0.8 A t , and the mass of a piece of metal of area 400 c m 2 and thickness 3 mm is 960 g .
Examples
In construction, understanding how the mass of materials varies with area and thickness is crucial for estimating the total weight of structures. For example, if you're building a roof using metal sheets, knowing the relationship M = k A t allows you to calculate the mass of the roofing material needed based on the area to be covered and the thickness of the sheets. This helps in planning the support structure and ensuring it can bear the load safely. This principle extends to various fields, including manufacturing, engineering, and even cooking, where scaling recipes or material quantities is essential.
