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Mathematics-Online problems:

Solution to the problem of the (previous) week

Problem:

Having redecorated two rooms, the decorator sent his customer the following invoice:
Work done Price per unit Sum
Ceiling painted white 7.00 EUR/m$ ^2$ 224.00 EUR
Walls painted colored 8.00 EUR/m$ ^2$ 640.00 EUR
Carpet fitted 9.50 EUR/m$ ^2$ 304.00 EUR
Skirting boards fixed 8.50 EUR/m 272.00 EUR
Sum   1440.00 EUR
    19 % VAT 273.60 EUR
Total   1713.60 EUR

\includegraphics[width=0.8\linewidth]{renovierung_en.eps}


What is the total volume of both rectangular rooms?
How long is the wall common to both rooms?
What are the external dimensions of the block of rooms, if walls, ceilings, and floors are 50 cm thick?

(Hint: Assume that it is an honest decorator; openings for doors and windows should not be taken into account.)

Answer:

Total volume: m$ ^3$
Length of wall: m
External dimensions: m x m x m

(The results should be filled in in descending order.)

Solution:


$\displaystyle (x_1+x_2)y$ $\displaystyle =$ $\displaystyle \frac{224}{7}\ $    (Ceiling)  
$\displaystyle 2x_1+2y+2x_2+2y$ $\displaystyle =$ $\displaystyle \frac{272}{8.5}\ $    (Skirting boards)  
$\displaystyle 2x_1z+2yz+2x_2z+2yz$ $\displaystyle =$ $\displaystyle \frac{640}{8}\ $    (Walls)  

$ \rightarrow$
$\displaystyle x_1+x_2$ $\displaystyle =$ $\displaystyle \frac{32}{y} \qquad \qquad (1)$  
$\displaystyle x_1+x_2+2y$ $\displaystyle =$ $\displaystyle 16 \qquad \qquad (2)$  
$\displaystyle (x_1+x_2+2y)z$ $\displaystyle =$ $\displaystyle 40\rightarrow\underline{\underline{z=2.5}}$  

From equations (1) and (2) we obtain

$\displaystyle \frac{32}{y}+2y=16\rightarrow 2y^2-16y+32=0 \rightarrow\underline{\underline{y=4}} $

$ \rightarrow$

$\displaystyle \underline{\underline{x_1+x_2=8}} $

Volume $ =(x_1+x_2)yz=80$m$ ^3$, common wall: 4m

External dimensions:

\includegraphics[width=0.6\textwidth]{aussen.eps},

$\displaystyle \rightarrow (x_1+x_2+1.5)\ \times\ (y+1)\ \times\ (z+1) \rightarrow \underline{\underline{9.5\ \times\ 5\ \times\ 3.5}} $

[problem of the week]