Asosiy tarkibga o'tish

4.9-bo'lim : Ko'proq optimallashtirish

  1. Biz oyna qurmoqchimiz, uning pastki qismi to'rtburchak, derazaning tepasi esa teng tomonli uchburchakdir. Agar bizda 75 dyuymli ramka materiali bo'lsa, derazaning o'lchamlari eng ko'p yorug'likni ta'minlaydi?
  2. Biz o'rtasi to'rtburchak, derazaning yuqori va pastki qismlari teng tomonli uchburchaklar bo'lgan oyna qurmoqchimiz. Agar bizda 4 fut ramka materiali bo'lsa, derazaning o'lchamlari qanday bo'lishi mumkin, bu eng yorug'likni beradi?
  3. Biz o'rtasi to'rtburchak, oynaning yuqori qismi yarim doira va pastki qismi teng tomonli uchburchak bo'lgan oyna qurmoqchimiz. Agar bizda 1500 sm ramka materiali bo'lsa, derazaning o'lchamlari qanday bo'lishi mumkin, bu eng yorug'likni beradi?
  4. Radiusi 5 bo‘lgan aylana ichiga chizilgan eng katta to‘rtburchakning maydonini aniqlang.
    A sketch of a rectangle inscribed in a circle. All four corners of the rectangle touch the circle and the rectangle is wider than it is tall.
  5. Poydevori \(x\) o'qida va yuqori ikki burchagi 16 radiusli yarim doira ustida joylashgan eng katta to'rtburchakning maydonini aniqlang.
    A sketch of a semicircle sitting on the x-axis with a rectangle inscribed in it. The base of the rectangle lies along the x-axis and the top two corners of the rectangle touch the semicircle.
  6. Bazasi \(x\) o'qida va yuqori ikki burchagi \(y = 4 - {x^2}\) bo'lgan eng katta to'rtburchakning maydonini aniqlang.
    A sketch of the downward opening parabola \(y = 4 - {x^2}\) with a rectangle inscribed under it. The base of the rectangle lies along the x-axis and the top two corners of the rectangle touch the parabola.
  7. \(\displaystyle \frac{{{x^2}}}{4} + \frac{{{y^2}}}{{36}} = 1\) da \(\left( {0,1} \right)\) ga eng yaqin nuqta(lar)ni toping.
  8. \(x = {y^2} - 8\) da \(\left( {5,0} \right)\) ga eng yaqin nuqta(lar)ni toping.
  9. \(y = 2 - {x^2}\) da \(\left( {0, - 3} \right)\) ga eng yaqin nuqta(lar)ni toping.
  10. 6 futlik sim ikki bo'lakka kesiladi. Bir bo'lak teng qirrali uchburchakka egiladi, ikkinchisi esa bir tomoni boshqa tomonining uzunligidan ikki baravar katta bo'lgan to'rtburchaklar shaklida egiladi. Ikkita raqam bilan o'ralgan maydonni minimallashtirish uchun simni qayerda kesish kerakligini aniqlang.
  11. 250 sm uzunlikdagi sim ikki qismga bo'linadi. Bir bo'lak teng qirrali uchburchakka egiladi, ikkinchisi esa aylana shaklida egiladi. Ikkala raqam bilan o'ralgan maydonni maksimal darajada oshirish uchun simni qayerda kesish kerakligini aniqlang.
  12. 250 sm uzunlikdagi sim ikki qismga bo'linadi. Bir bo'lak teng qirrali uchburchakka egiladi, ikkinchisi esa aylana shaklida egiladi. Ikkita raqam bilan o'ralgan maydonni minimallashtirish uchun simni qayerda kesish kerakligini aniqlang.
  13. 4 m uzunlikdagi sim ikki qismga bo'linadi. Bir bo'lak aylana shaklida egiladi, ikkinchisi esa bir tomoni boshqa tomonning uzunligidan uch baravar uzun bo'lgan to'rtburchaklar shaklida egiladi. Ikkala raqam bilan o'ralgan maydonni maksimal darajada oshirish uchun simni qayerda kesish kerakligini aniqlang.
  14. \(\left( { - 4,1} \right)\) nuqtasi orqali o'tadigan chiziq \(x\) o'qi va \(y\) o'qi 2 bo'lgan to'g'ri burchakli uchburchak hosil qiladi.nd kvadrant. Ushbu uchburchakning maydonini minimallashtiradigan chiziq tenglamasini aniqlang.
    A sketch of the triangle described in the problem. A line rises from the lower left to the upper right, passing through the point \(\left( { - 4,1} \right)\), which is marked on the line. The line together with the x-axis and the y-axis forms a right triangle in the second quadrant and the inside of the triangle is shaded and labeled “Minimize This Area”.
  15. \(\left( {3,3} \right)\) nuqtadan o'tadigan chiziq \(x\) o'qi va \(y\) o'qi bilan to'g'ri burchakli uchburchak hosil qiladi.st kvadrant. Ushbu uchburchakning maydonini minimallashtiradigan chiziq tenglamasini aniqlang.
    A sketch of the triangle described in the problem. A line falls from the upper left to the lower right, passing through the point \(\left( {3,3} \right)\), which is marked on the line. The line together with the x-axis and the y-axis forms a right triangle in the first quadrant and the inside of the triangle is shaded and labeled “Minimize This Area”.
  16. Quvur bo'lagi 14 fut kenglikdagi koridor bo'ylab olib ketilmoqda. Yo'lakning oxirida to'g'ri burchakli burilish mavjud va yo'lak 6 fut kengligida torayadi. Koridorda burilish bo'ylab ko'tarilishi mumkin bo'lgan eng uzun quvur (har doim gorizontal holatda) nima?
    A sketch of the hallway described in the problem, seen from above. The wide hallway is 14 feet across and runs vertically, the narrow hallway is 6 feet across and runs horizontally, and they meet at a right-angled turn. The pipe is drawn as a straight line lying diagonally across the inside corner of the turn, touching the outer wall of each hallway and resting against the inner corner.
  17. Quvur bo'lagi 9 fut kenglikdagi koridor bo'ylab olib ketilmoqda. Yo'lakning oxirida to'g'ri burchakli burilish mavjud va koridor kengligi 25 metrgacha kengayadi. Koridorda burilish bo'ylab ko'tarilishi mumkin bo'lgan eng uzun quvur (har doim gorizontal holatda) nima?
    A sketch of the hallway described in the problem, seen from above. The first hallway is 9 feet across and runs vertically, the second hallway is 25 feet across and runs horizontally, and they meet at a right-angled turn. The pipe is drawn as a straight line lying diagonally across the inside corner of the turn, touching the outer wall of each hallway and resting against the inner corner.
  18. Birining balandligi 15 metr va balandligi 10 metr bo'lgan ikkita ustun bir-biridan 40 metr masofada joylashgan. Har bir qutbning yuqori qismiga uzunlikdagi sim bog'langan va u ikki qutb o'rtasida bir joyda erga qo'yilgan. Minimal sim miqdori ishlatilishi uchun simni qaerga qo'yish kerak?
    A sketch of the two poles and the wire. The 15 meter pole is on the left, the 10 meter pole is on the right and the ground between them is marked as 40 meters. The wire runs from the top of the taller pole down to a stake on the ground between the poles and then back up to the top of the shorter pole, forming a “V” shape.
  19. Ikkita ustun, biri 2 fut va biri 5 fut balandlikda, bir-biridan 3 fut masofada joylashgan. Har bir qutbning yuqori qismiga uzunlikdagi sim bog'langan va u ikki qutb o'rtasida bir joyda erga qo'yilgan. Minimal sim miqdori ishlatilishi uchun simni qaerga qo'yish kerak?
    A sketch of the two poles and the wire. The 2 foot pole is on the left, the 5 foot pole is on the right and the ground between them is marked as 3 feet. The wire runs from the top of the shorter pole down to a stake on the ground between the poles and then back up to the top of the taller pole, forming a “V” shape.
  20. Birining balandligi 15 metr va balandligi 10 metr bo'lgan ikkita ustun bir-biridan 40 metr masofada joylashgan. Har bir qutbning yuqori qismiga uzunlikdagi sim bog'langan va u ikki qutb o'rtasida bir joyda erga qo'yilgan. Qoziqdagi ikkita simdan hosil bo'lgan burchak maksimal bo'lishi uchun simni qayerga qo'yish kerak?
    A sketch of the two poles and the wire. The 15 meter pole is on the left, the 10 meter pole is on the right and the ground between them is marked as 40 meters. The wire runs from the top of the taller pole down to a stake on the ground and then back up to the top of the shorter pole, and the angle formed by the two pieces of wire at the stake is labeled \(\theta \).
  21. Biri 34 dyuym va 17 dyuym bo'lgan ikkita qutb bir-biridan 3 fut masofada joylashgan. Har bir qutbning yuqori qismiga uzunlikdagi sim bog'langan va u ikki qutb o'rtasida bir joyda erga qo'yilgan. Qoziqdagi ikkita simdan hosil bo'lgan burchak maksimal bo'lishi uchun simni qayerga qo'yish kerak?
    A sketch of the two poles and the wire. The 34 inch pole is on the left, the 17 inch pole is on the right and the ground between them is marked as 3 feet. The wire runs from the top of the taller pole down to a stake on the ground and then back up to the top of the shorter pole, and the angle formed by the two pieces of wire at the stake is labeled \(\theta \).
  22. Quyidagi rasmda ko'rsatilganidek, suvni ushlab turish uchun oluk hosil bo'lishi kerak. Chuqur ushlab turadigan suv miqdorini maksimal darajada oshiradigan \(\theta \) burchagini aniqlang.
    The base of the trough is shown and labeled “30 in”. There is also a lighter gray shaded line that extends out from the right and left side of the base of the trough. The two sides are shown and also labeled “30 in”. The angle that each side makes with the gray line underneath is given as $\theta$.
  23. Quyidagi rasmda ko'rsatilganidek, suvni ushlab turish uchun oluk hosil bo'lishi kerak. Chuqur ushlab turadigan suv miqdorini maksimal darajada oshiradigan \(\theta \) burchagini aniqlang.
    The base of the trough is shown and labeled “4 m”. There is also a lighter gray shaded line that extends out from the right and left side of the base of the trough. The two sides are shown and labeled “1 m”. The angle that each side makes with the gray line underneath is given as $\theta$.