13.153. A plot in the shape of a rectangle with a diagonal of 185 m was allocated for a sports field. During construction, the length of each side was reduced by 4 m. The rectangular shape was maintained, but the area was reduced by \(1012 \mathrm{~m}^{2}\). What are the actual dimensions of the sports field?

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-876.jpg?height=354&width=502&top_left_y=363&top_left_x=747)

Fig. 13.10
Solution.

Let $x$ be the length and $y$ be the width of the plot. By the Pythagorean theorem, $x^{2}+y^{2}=185^{2}$. Additionally, $(x-4)(y-4)=x y-1012$, from which $x y-4(x+y)=$ $=x y-1028 ; x+y=257 ; y=257-x$. Solving the quadratic equation $x^{2}+(257-x)^{2}-185^{2}=0$, we find $x_{1}=104, x_{2}=153$.

Answer: $100 \times 149$ m.

13.154.3a 1 kg of one product and 10 kg of another cost 20 rubles. If, due to seasonal price changes, the first product increases in price by $15 \%$, and the second decreases in price by $25 \%$, then the same amount of these products will cost 18 rubles 20 kopecks. What is the cost of 1 kg of each product?

Solution.

Let $x$ rubles be the price of one product, and $y$ rubles be the price of the other. According to the problem, $x+10 y=20$. With the seasonal price changes, 1 kg of the first product will cost $1.15 x$ rubles, and 1 kg of the second product will cost $0.75 y$ rubles. Then $1.15 x+10 \cdot 0.75 y=18.2$. Solving the system $\left\{\begin{array}{l}1.15 x+0.75 \cdot 10 y=18.2, \\ x+10 y=20,\end{array}\right.$ we find $x=8$ rubles, $y=1.2$ rubles.

Answer: 8 rubles and 1 ruble 20 kopecks.