A smart grid feasibility study categorized 1000 commercial buildings based on their adopti… — State PSC Exam

A smart grid feasibility study categorized $1000$ commercial buildings based on their adoption of three energy efficiency upgrades: Smart Lighting ( $L$ ), HVAC Optimization ( $H$ ), and Automated Shading ( $A$ ).

The survey revealed the following:
- $150$ buildings have not adopted any of these three upgrades.
- The number of buildings that have adopted exactly one type of upgrade is identical across all three categories.
- The number of buildings with both $L$ and $H$ , but not $A$ , is twice the number of buildings with all three upgrades.
- The number of buildings with both $H$ and $A$ , but not $L$ , is three times the number of buildings with all three upgrades.
- The number of buildings with both $L$ and $A$ , but not $H$ , is equal to the number of buildings with all three upgrades.
- The total number of buildings that have adopted HVAC Optimization ( $H$ ) is $430$ .

Based on this data, what is the total number of buildings that have adopted exactly two types of upgrades?

240Answer
B
280
C
330
D
40

Answer

240 buildings

By defining the 'exactly one' regions as

x

and the 'all three' region as

y

, we can model the 'exactly two' regions as

2y

3y

, and

y

. This gives us two linear equations based on the given totals:

3x + 7y = 850

(from the union of

1000 - 150

) and

x + 6y = 430

(from the components of set

H

). Solving this system yields

y = 40

. The question asks for the total number of buildings with exactly two upgrades, which is

2y + 3y + y = 6y

. Multiplying

6 \times 40

gives

240

Step-by-Step Solution

Determine the total number of buildings with at least one upgrade.

Total Union

= 1000 - 150 = 850

buildings.

Subtracting the buildings with none of the upgrades from the total surveyed gives the union of the three sets.

Define algebraic variables for the distinct regions of the Venn diagram.

Let the number of buildings with exactly one upgrade in any category be

x

. Let the number of buildings with all three upgrades be

y

Parameterizing the unknown regions allows us to express the given relationships algebraically.

Express the regions representing exactly two upgrades in terms of

y

n(L \cap H \text{ only}) = 2y

n(H \cap A \text{ only}) = 3y

, and

n(L \cap A \text{ only}) = y

. The total for exactly two upgrades is

2y + 3y + y = 6y

Translating the comparative statements in the prompt into exact algebraic expressions.

Formulate a system of linear equations using the Total Union and Total H.

Total Union equation:

3x + (2y + 3y + y) + y = 850 \Rightarrow 3x + 7y = 850

. Total H equation:

x + 2y + 3y + y = 430 \Rightarrow x + 6y = 430

The union includes all distinct regions. The Total H set includes H only, all intersections involving H, and the central intersection.

Solve the system of equations to find

y

and calculate the final target (

6y

From the second equation,

x = 430 - 6y

. Substituting into the first:

3(430 - 6y) + 7y = 850 \Rightarrow 1290 - 18y + 7y = 850 \Rightarrow 11y = 440 \Rightarrow y = 40

. The total for exactly two is

6(40) = 240

This isolates

y

, giving the base unit needed to find the number of buildings with exactly two upgrades.

Key Concept

Algebraic Formulation of Multi-Set Logical Venn Diagrams

Question

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