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Perhatikan gambar berikut! Pada gambar, terdapat tabung dan kerucut dengan alas berimpit dan tingginya sama. Tentukan: a. volume tabung b. volume kerucut c. volume tanung di luar kerucut

Pertanyaan

Perhatikan gambar berikut!

Pada gambar, terdapat tabung dan kerucut dengan alas berimpit dan tingginya sama. Tentukan:

a. volume tabung

b. volume kerucut

c. volume tanung di luar kerucut

Pembahasan Soal:

a. Volume tabung

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell V subscript t end cell equals cell πr squared straight t end cell row blank equals cell 22 over 7 cross times 7 squared cross times 27 end cell row blank equals cell 4.158 space cm cubed end cell end table end style 

b. Volume kerucut

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell V subscript k end cell equals cell 1 third πr squared straight t end cell row blank equals cell 1 third cross times 22 over 7 cross times 7 squared cross times 27 end cell row blank equals cell 1.386 space cm cubed end cell end table end style 

c. Volume tabung di luar kerucut

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row V equals cell V subscript t minus V subscript k end cell row blank equals cell 4.158 minus 1.386 end cell row blank equals cell 2.772 space cm cubed end cell end table end style

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

H. Janatu

Mahasiswa/Alumni Universitas Riau

Terakhir diupdate 30 Maret 2021

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Pertanyaan yang serupa

Perhatikan gambar dibawah ini! Sebuah tabung dan kerucut yang panjang diameternya  adalah 14 cm. Hitunglah volume bangun!

Pembahasan Soal:

Diketahui:

table attributes columnalign right center left columnspacing 0px end attributes row d equals cell 14 space cm end cell row r equals cell 1 half cross times d end cell row blank equals cell fraction numerator 1 over denominator up diagonal strike 2 end fraction cross times up diagonal strike 14 to the power of 7 end strike end cell row blank equals cell 7 space cm end cell row cell t subscript t a b u n g end subscript end cell equals cell 20 space cm end cell row cell t subscript t o t a l end subscript end cell equals cell 44 space cm end cell row cell t subscript k e r u c u t end subscript end cell equals cell 44 minus 20 end cell row blank equals cell 24 space cm end cell end table

Ditanya: Volume bangun ?

Voilume bangun terssebut adalah gabungan volume tabung dan volume kerucut.

Volume bangun dapat dicari dengan cara 

table attributes columnalign right center left columnspacing 0px end attributes row cell Volume space bangun end cell equals cell Volume space tabung plus Volume space kerucut end cell row blank equals cell left parenthesis straight pi times r squared times t subscript t a b u n g end subscript right parenthesis space plus open parentheses 1 third times straight pi times r squared times t subscript k e r u c u t end subscript close parentheses end cell row blank equals cell open parentheses fraction numerator 22 over denominator up diagonal strike 7 end fraction times up diagonal strike 7 to the power of 2 to the power of space 7 end exponent end exponent end strike times 20 close parentheses plus open parentheses fraction numerator 1 over denominator up diagonal strike 3 end fraction times fraction numerator 22 over denominator up diagonal strike 7 end fraction times up diagonal strike 7 to the power of 2 to the power of 7 end exponent end strike times up diagonal strike 24 to the power of 8 end strike close parentheses end cell row blank equals cell left parenthesis 22 cross times 7 cross times 20 right parenthesis plus left parenthesis 22 cross times 7 cross times 8 right parenthesis end cell row blank equals cell 3.080 space plus space 1232 end cell row blank equals cell 4.312 space cm cubed end cell end table

Jadi, volume bangun tersebut adalah 4.312 cm3.

0

Roboguru

Perhatikan kembali gambar pada Soal nomor 12 . a. Tentukan perbandingan volume A dengan volume B. b. Tentukan perbandingan volume dengan volume C . c. Tentukan perbandingan volume  dengan volume D....

Pembahasan Soal:

Dalam soal ini ada 4 buah kerucut dengan alas di A,B,C, dan D. Kita mengetahui bahwa:

begin mathsize 14px style t inggi space kerucut space A colon B colon C colon D equals 4 x colon 3 x colon 2 x colon x space atau space bisa space disederhanakan space menjadi space 4 colon 3 colon 2 colon 1 end style. Dengan menggunkan prinsip kesebanguan, dapat diketahui bahwa perbandingan antara jari-jari dan juga tinggi tiap kerucut adalah sama. Maka perbandingan begin mathsize 14px style j a r i minus j a r i space kerucut space A colon B colon C colon D equals 4 colon 3 colon 2 colon 1 space atau space sama space juga space dengan space 4 r colon 3 r colon 2 r colon 1 r end style . Dengan informasi jari-jari dan tinggi kerucut maka kita dapat menyelesaikan soal ini.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell a. space Volume space A end cell equals cell Volume space Kerucut subscript A minus Volume space kerucut subscript B end cell row cell space space space space space V o l u m e space B end cell equals cell Volume space Kerucut subscript B minus Volume space kerucut subscript C end cell row blank blank cell space space space space space end cell end table end style 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator Volume space A over denominator Volume space B end fraction end cell equals cell fraction numerator begin display style 1 third end style pi open parentheses 4 r close parentheses squared 4 x minus 1 third pi open parentheses 3 r close parentheses squared 3 x over denominator 1 third pi open parentheses 3 r close parentheses squared 3 x minus 1 third pi open parentheses 2 r close parentheses squared 2 x end fraction end cell row blank equals cell fraction numerator begin display style 1 third pi times 16 r squared times 4 x minus 1 third pi times 9 r squared times 3 x end style over denominator 1 third pi times 9 r squared times 3 x minus 1 third pi times 4 r squared times 2 x end fraction end cell row blank equals cell fraction numerator 1 third pi r squared x left parenthesis 64 minus 27 right parenthesis over denominator 1 third pi r squared x left parenthesis 27 minus 8 right parenthesis end fraction end cell row blank equals cell 37 over 19 end cell row cell Volume space A colon Volume space B end cell equals cell 37 colon 19 end cell row blank blank blank end table end style  

 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell b. space Volume space B end cell equals cell Volume space Kerucut blank subscript B minus Volume space kerucut blank subscript C space space space space space end cell row cell space space space V o l u m e space C end cell equals cell Volume space Kerucut blank subscript C minus Volume space kerucut blank subscript D end cell end table end style   

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator Volume space B over denominator Volume space C end fraction end cell equals cell fraction numerator begin display style 1 third pi open parentheses 3 r close parentheses squared 3 x minus 1 third pi open parentheses 2 r close parentheses squared 2 x end style over denominator 1 third pi left parenthesis 2 r right parenthesis squared 2 x minus 1 third pi r squared x end fraction end cell row blank equals cell fraction numerator begin display style 1 third pi times 9 r squared times 3 x minus 1 third pi times 4 r squared times 2 x end style over denominator 1 third pi times 4 r squared times 2 x minus 1 third pi times r squared times x end fraction end cell row blank equals cell fraction numerator 1 third pi r squared x left parenthesis 27 minus 8 right parenthesis over denominator 1 third pi r squared x left parenthesis 8 minus 1 right parenthesis end fraction end cell row blank equals cell 19 over 7 end cell row cell Volume space B colon Volume space C end cell equals cell 19 colon 7 end cell end table end style 

 

Error converting from MathML to accessible text. 

  begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator Volume space C over denominator Volume space D end fraction end cell equals cell fraction numerator begin display style fraction numerator begin display style 1 end style over denominator begin display style 3 end style end fraction pi left parenthesis 2 r right parenthesis squared 2 x minus fraction numerator begin display style 1 end style over denominator begin display style 3 end style end fraction pi r squared x end style over denominator 1 third pi r squared x end fraction end cell row blank equals cell fraction numerator begin display style 1 third pi times 4 r squared times 2 x minus fraction numerator begin display style 1 end style over denominator begin display style 3 end style end fraction pi r squared x end style over denominator 1 third pi times r squared times x end fraction end cell row blank equals cell fraction numerator 1 third pi r squared x left parenthesis 8 minus 1 right parenthesis over denominator 1 third pi r squared x end fraction end cell row blank equals cell 7 over 1 end cell row cell Volume space C colon Volume space D end cell equals cell 7 colon 1 end cell end table end style   

0

Roboguru

Untuk soal 7 perhatikan tabel di bawah ini. Tentukan luas permukaan dan volume tiap-tiap bangun.

Pembahasan Soal:

 begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight a. space Luas space Permukaan end cell equals cell luas space alas space tabung plus luas space selimut space tabung plus luas space selimut space kerucut end cell row blank equals cell pi r squared plus 2 pi r t plus pi r s end cell row blank equals cell pi r left parenthesis r plus 2 t plus s right parenthesis end cell end table end style      

            Untuk mendapatkan nilai begin mathsize 14px style s end style atau garis pelukis digunakan konsep phytagoras pada kerucut sebagai berikut: 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell s squared end cell equals cell t squared plus r squared end cell row s equals cell square root of t squared plus r squared end root end cell end table end style  

     begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row volume equals cell volume space tabung plus volume space kerucut end cell row blank equals cell pi r to the power of 2 space end exponent t space plus 1 third pi r squared t end cell row blank equals cell 4 over 3 pi r squared t end cell end table end style  

 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight b. space Luas space permukaan end cell equals cell 2 cross times space luas space selimut space kerucut space plus luas space selimut space tabung end cell row blank equals cell 2 cross times pi r s plus 2 pi r t end cell row blank equals cell 2 pi r s plus 2 pi r t end cell row blank equals cell 2 pi r open parentheses s plus t close parentheses end cell end table end style 

 

    begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row Volume equals cell 2 cross times volume space kerucut plus volume space tabung end cell row blank equals cell 2 cross times 1 third pi r squared t plus pi r squared t end cell row blank equals cell 2 over 3 pi r squared t plus pi r squared t space end cell row blank equals cell left parenthesis 2 over 3 plus 1 right parenthesis pi r squared t end cell row blank equals cell 5 over 3 pi r squared t end cell end table end style  

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight c. space Luas space permukaan end cell equals cell 1 half luas space permukaan space bola space plus space luas space selimut space kerucut end cell row blank equals cell 1 half cross times 4 pi r squared plus pi r s end cell row blank equals cell 2 pi r squared plus pi r s end cell row blank equals cell pi r open parentheses 2 r plus s close parentheses end cell end table end style 

 

     begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row Volume equals cell 1 half volume space bola plus volume space kerucut end cell row blank equals cell 1 half cross times 4 over 3 pi r cubed plus 1 third pi r squared t end cell row blank equals cell 2 over 3 pi r cubed plus 1 third pi r squared t end cell row blank equals cell 1 third pi r squared left parenthesis 2 r plus t right parenthesis end cell end table end style   

 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell d. Luas space Permukaan end cell equals cell 1 half Luas space permukaan space bola plus luas space selimut space tabung plus luas space tutup space tabung end cell row blank equals cell 1 half cross times 4 pi r squared plus 2 pi r t plus pi r squared end cell row blank equals cell 2 pi r squared plus 2 pi r t plus pi r squared end cell row blank equals cell 3 pi r squared plus 2 pi r t end cell row blank equals cell pi r open parentheses 3 r plus 2 t close parentheses end cell end table end style 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell Volume space end cell equals cell 1 half Volume space bola plus volume space tabung end cell row blank equals cell 1 half cross times 4 over 3 pi r cubed plus pi r squared t end cell row blank equals cell 2 over 3 pi r cubed plus pi r squared t end cell row blank equals cell pi r squared open parentheses 2 over 3 r plus t close parentheses end cell end table end style 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight e. space Luas space Permukaan space end cell equals cell 1 half space luas space permukaan space bola space plus space luas space selimut space tabung space plus luas space selimut space kerucut end cell row blank equals cell 1 half cross times 4 pi r squared plus 2 pi r t plus pi r s end cell row blank equals cell 2 pi r squared plus 2 pi r t plus pi r s end cell row blank equals cell pi r open parentheses 2 r plus 2 t plus s close parentheses end cell end table end style 

 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row Volume equals cell 1 half volume space bola plus volume space tabung plus volume space kerucut end cell row blank equals cell 1 half cross times 4 over 3 pi r cubed plus pi r squared t plus 1 third pi r squared t end cell row blank equals cell 2 over 3 pi r cubed plus pi r squared t plus 1 third pi r squared t end cell row blank equals cell pi r squared left parenthesis 2 over 3 r plus 4 over 3 t right parenthesis end cell row blank equals cell 2 over 3 pi r squared left parenthesis r plus 2 t right parenthesis end cell end table end style   

 

undefined 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row volume equals cell 2 cross times 1 half volume space bola plus volume space tabung end cell row blank equals cell 4 over 3 pi r cubed plus pi r squared t end cell row blank equals cell pi r squared open parentheses 4 over 3 r plus t close parentheses end cell end table end style

 

 

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Roboguru

Volume gabungan bangun ruang di atas adalah ...

Pembahasan Soal:

Volume bangun ruang gabungan pada soal dapat dihitung dengan menjumlahkan volume tabung dan volume kerucut. Rumus volume tabung dan volume kerucut adalah sebagai berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell straight V subscript Tabung end cell equals cell pi r squared t end cell row cell straight V subscript Kerucut end cell equals cell 1 third pi r squared t end cell end table 

Dimana r adalah jari-jari, dan t adalah tinggi. Jari-jari tabung sama dengan jari-jari kerucut yaitu 14 over 2 space cm. Sedangkan tinggi kerucut yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell t subscript Kerucut end cell equals cell t subscript Gabungan minus t subscript Tabung end cell row blank equals cell 18 minus 12 end cell row blank equals cell 6 space cm end cell end table 

Volume tabung:

table attributes columnalign right center left columnspacing 0px end attributes row cell straight V subscript Tabung end cell equals cell pi r squared t subscript Tabung end cell row blank equals cell 22 over 7 cross times open parentheses 14 over 2 close parentheses squared cross times 12 end cell row blank equals cell 22 over 7 cross times open parentheses 7 close parentheses squared cross times 12 end cell row blank equals cell 22 cross times 7 cross times 12 end cell row blank equals cell 1.848 space cm cubed end cell end table 

Volume Kerucut:

table attributes columnalign right center left columnspacing 0px end attributes row cell straight V subscript Kerucut end cell equals cell 1 third pi r squared t subscript Kerucut end cell row blank equals cell 1 third cross times 22 over 7 cross times open parentheses 14 over 2 close parentheses squared cross times 6 end cell row blank equals cell 22 over 7 cross times open parentheses 7 close parentheses squared cross times 1 third cross times 6 end cell row blank equals cell 22 cross times 7 cross times 2 end cell row blank equals cell 308 space cm cubed end cell end table 

Sehingga, volume bangun ruang gabungan tersebut yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell straight V subscript Gabungan end cell equals cell straight V subscript Tabung plus straight V subscript Kerucut end cell row blank equals cell 1.848 plus 308 end cell row blank equals cell 2.156 space cm cubed end cell end table 

Dengan demikian, volume bangun ruang gabungan tersebut adalah 2.156 space cm cubed.

Jadi, jawaban yang benar adalah D.

0

Roboguru

Untuk tiap pasangan bangun ruang yang sebangun, hitung volume yang belum diketahui. a.  b.  c. . Dari jawaban  14a dan 14b , kesimpulan apa yang dapat diperoleh?

Pembahasan Soal:

a. Untuk menyelesaikan soal di atas kita menggunakan konsep kesebangunan. Informasi yang sudah kita miliki adalah:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell v subscript 1 end cell equals cell 12 pi space cm cubed end cell row cell s subscript 1 end cell equals cell 5 space cm end cell row cell s subscript 2 end cell equals cell 15 space cm end cell end table end style

 

Dengan konsep kesebangunan maka kita akan mendapatkan nilai begin mathsize 14px style r space dan space t end style 

begin mathsize 14px style r subscript 1 over r subscript 2 equals t subscript 1 over t subscript 2 equals s subscript 1 over s subscript 2 equals 5 over 15 equals 1 third  m a k a space d i p e r o l e h space r subscript 2 equals 3 r subscript 1 space end subscript d a n space space t subscript 2 equals 3 t subscript 1 space end subscript end style 

Maka kita dapat menghitung volume begin mathsize 14px style v subscript 2 end style dengan menggunakan rumus volume kerucut

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell v subscript 2 end cell equals cell 1 third pi open parentheses r subscript 2 close parentheses squared t subscript 2 end cell row blank equals cell 1 third pi open parentheses 3 r subscript 1 close parentheses squared open parentheses 3 t subscript 1 close parentheses end cell row blank equals cell open parentheses 9 times 3 close parentheses 1 third pi r subscript 1 squared t subscript 1 end cell row blank equals cell 27 times v subscript 1 end cell row blank equals cell 27 times 12 pi end cell row blank equals cell 324 pi space cm cubed end cell row blank blank blank end table end style  

 

b. Konsep yang sama dengan no a diterapkan pada soal b. Pada soal tertulis begin mathsize 14px style L equals 200 pi space cm cubed space Karena space satuannya space kubik space maka space nilai italic space L space ini space adalah space volume comma space bukan space luas space permukaan comma space mungkin space maksudnya space adala space Volume space bangun space L comma space sehingga space kita space anggap space saja space L space itu space adalah space straight v subscript 1 end style

begin mathsize 14px style r subscript 1 over r subscript 2 equals t subscript 1 over t subscript 2 equals 10 over 5 equals 2 over 1 space m a k a space d i p e r o l e h space r subscript 2 equals 1 half r subscript 1 space end subscript d a n space space t subscript 2 equals 1 half t subscript 1 space space end subscript end style 

Maka kita dapat menghitung volume begin mathsize 14px style v subscript 2 end style dengan menggunakan rumus volume tabung

Error converting from MathML to accessible text. 

 

c. Kesimpulannya adalah perbandingan volume kerucut dan tabung yang sebangun adalah sebesar pangkat 3 dari besar perbandingannya jari-jari atau tingginya atau garis pelukisnya. begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell v subscript 2 end cell equals cell open parentheses r subscript 2 over r subscript 1 close parentheses cubed v subscript 1 end cell row cell v subscript 2 end cell equals cell open parentheses t subscript 2 over t subscript 1 close parentheses cubed v subscript 1 end cell row cell v subscript 2 end cell equals cell open parentheses s subscript 2 over s subscript 1 close parentheses cubed v subscript 1 space space space space open parentheses hanya space pada space kerucut close parentheses end cell row blank blank blank end table end style 

 

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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