Missing Measurements According to Area: Decomposable and Truncated Solids
To successfully calculate a missing measure in a decomposable or truncated solid using the area, the formulas for the area of plane figures are useful. The approach is more complex than the one for simple solids like prisms and pyramids, because the number of plane figures used in the construction of a decomposable solid is greater. To understand the examples in this concept sheet better, the following is an example of the steps used.
Rule
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Determine the area formula to use.
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Identify the missing measurement(s), using a variable.
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Associate the area of each figure with a numerical value or an algebraic expression.
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Create an equation based on the context.
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Solve the equation.
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Interpret the answer according to the context.
Finding a Missing Measurement from a First-Degree Equation
To increase marketplace visibility, a company working in the pet supply industry wants to market a new toy for dogs. To attract customers to the new product, it plans to cover the toy with a product that has a smell and taste dogs love.
After a few calls, the company finds a supplier willing to sell a product for $0.02/cm2 of the surface area to be covered. To maximize profits, the company will invest $9.20 to cover each toy with the product.
To respect the profit margin, what should be the toy’s length?
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In general, it is a matter of properly identifying the unknown and using the appropriate formulas to create an equation that summarizes the situation. Once the equation is solved, we simply interpret the answer.
Finding a Missing Measurement from a Second-Degree Equation
To find the missing measurements according to the area of complex solids with second-degree equations, the procedure is similar to the one involving first-degree equations. However, the methods for solving the problem in step 5 will be a little different. In general, we can rely on factorization by the product-sum technique or the quadratic formula to solve such an equation.
A company specializes in the manufacture of crampons. To meet product demand, it needs to build a crampon that resembles the following.
The crampon also must meet the following specifications.
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The base’s length must measure 10 mm more than the cone’s apothem.
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The base’s width must measure 4 mm less than the cone’s apothem.
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The prism’s height must measure exactly 6 mm.
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The measure of the cone’s radius must be exactly 2 mm.
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The total area of a crampon must be 600 mm2.
What are the precise measurements of each of the crampon’s dimensions?
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Finding a Missing Measurement from a Truncated Solid
Due to its particular construction, a truncated solid is often associated with a decomposable solid. In fact, it is a solid with a portion removed. In some cases, it is necessary to establish a proportion between the measurements of the complete solid and the truncated part. In other cases, it’s enough to analyze each of the solid’s faces and decompose them appropriately to find polygons with which we are more familiar. We can apply the first method to truncated cones, as in the following example.
The following rubber stopper has a total area of 105.61 cm2. What is the height of the stopper if the diameter of the small circle is 4 cm and the large one is 6 cm? The stopper is a truncated cone where the 2 bases are parallel circles.
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The next example shows a situation with a truncated prism.
To renovate an apartment building, the owner decides to replace the exterior siding and change the roof’s structure. Instead of a flat roof, he wants a roof sloped on two sides.
To cover the cost of the new siding, the budget is $30 000. The construction material costs $27.70/m2. What should be the height of the building’s new roof with two sloped sides?
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Finding a Missing Measurement from a Non-Convex Decomposable Solid
Since we are working in three dimensions, there are occasionally missing or hidden sections of the solid. Regardless, we can generally break down a non-convex (concave) solid into the different plane figures of which it consists.
The approach could be similar to the following.
As a hobby, Natasha makes customized cakes. To respect her recent client’s budget, she has $11 remaining to invest on icing for the cake.
If it costs her $0.50/dm2 for the necessary ingredients, what should be the height of the cake?
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