13.371. A section of an embankment for a railway is being constructed, 100 m long, with a cross-section in the form of an isosceles trapezoid with a lower base of 5 m, an upper base of no less than 2 m, and a slope angle of $45^{\circ}$. What height $h$ should this embankment have so that the volume of earthworks is no less than $400 \mathrm{M}^{3}$ but no more than $500 \mathrm{M}^{3}$?

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-1225.jpg?height=270&width=729&top_left_y=1514&top_left_x=322)

Fig. 13.21
## Solution.

From Fig. 13.21, we have $x=\frac{a-b}{2} ; h=x \operatorname{tg} 45^{\circ}=\frac{a-b}{2}$. The area of trapezoid $A B C D S=\frac{a+b}{2} h=\frac{a^{2}-b^{2}}{4}$.

According to the condition $400 \leq \frac{a^{2}-b^{2}}{4} \cdot 100 \leq 500$, from which $\sqrt{5} \leq b \leq 3$ (m). Then $\frac{a-3}{2} \leq h \leq \frac{a-\sqrt{5}}{2}$ (m) or $1 \leq h \leq \frac{5-\sqrt{5}}{2}$ (m).

Answer: $1 \leq h \leq \frac{5-\sqrt{5}}{2} \mathrm{M}$.
Solutions to Chapter 2. Identical Transformations of
Algebraic Expressions ..... 3
Solutions to Chapter 3. Identical Transformations of
Trigonometric Expressions ..... 131
Solutions to Chapter 4. Progressions ..... 289
Solutions to Chapter 6. Algebraic Equations ..... 312
Solutions to Chapter 7. Logarithms. Exponential
and Logarithmic Equations ..... 398
Solutions to Chapter 8. Trigonometric Equations ..... 468
Solutions to Chapter 9. Inequalities ..... 629
Solutions to Chapter 10. Problems in Plane Geometry ..... 705
Solutions to Chapter 11. Problems in Solid Geometry ..... 817
Solutions to Chapter 12. Problems in Geometry with the Application of
Trigonometry ..... 891
Solutions to Chapter 13. Application of Equations to Solving
Problems ..... 1153

# AUTHOR COLLECTIVE

Egerov Viktor Konstantinovich, Zaytsev Vladimir Valentinovich, Kordemsky Boris Anastasievich, Maslova Tamara Nikolaevna, Orlovskaya Iraida Fedorovna, Pozoysky Roman Isaevich, Ryakhovskaya Galina Sergeevna, Skanavi Mark Ivanovich, Sukhodsky Andrey Matveevich, Fedorova Nina Mikhailovna

## CREATIVE COLLECTIVE

Karpuk Andrey Andreyevich - Professor of the Department of Higher Mathematics at the Belarusian State University of Information and Radioelectronics

Zhevnjak Rostislav Mikhailovich - Professor of the Department of Higher Mathematics at the Belarusian State University of Information and Radioelectronics

Ermolitsky Alexander Alexandrovich - Candidate of Physical and Mathematical Sciences

# COMPLETE COLLECTION OF SOLUTIONS TO PROBLEMS IN MATHEMATICS

for applicants to higher education institutions

## Group B

## Edited by M. I. Skanavi

Signed for printing from ready transparencies of the customer on 18.10.2011. Format $60 \times 90^{\prime} / 16$ : Font "Times". Offset printing. Newspaper paper. Usual printed sheets 77,00. Additional print run 3000 copies. Order 1048.

All-Russian Classifier of Products OK-005-93, vol. $2 ; 953005$ - educational literature

LLC "Publishing House 'World and Education'". 115193, Moscow, 5th Kozhukhovskaya St., 13, bld. 1. Tel./fax: (495) 742-43-51, 742-43-54.

E-mail: mir-obrazovanie@onyx.ru

LLC "Publishing House ASTRAL". 129085, Moscow, Oliminskogo Proezd, 13, bld. 1.

The publication was carried out with the technical assistance of LLC "Publishing House AST".

Published with the participation of LLC "Harvest", license No. 02330/0494377 dated 16.03.2009. Republic of Belarus, 220013, Minsk, Kulman St., 1, bld. 3, fl. 4, off. 42. E-mail of the editorial office: harvest@anitex.by

OAO "Polygraphkombinat im. Ya. Kolas".

LP No. $02330 / 0150496$ or 11.03 .2009 .

Republic of Belarus, 220600, Minsk, Krasnaya St., 23.