Um vinho tinto é trazido da adega, que está fria com
, e deixado em uma sala com temperatura de
. Se forem necessários 10 minutos para o vinha alcançar
, quando sua temperatura atingirá
?
a. aproximadamente 22,5 minutos.
b. aproximadamente 18 minutos.
c. aproximadamente 20,7 minutos.
d. aproximadamente 15 minutos.
e. aproximadamente 19,7 minutos.
a. aproximadamente 22,5 minutos.
b. aproximadamente 18 minutos.
c. aproximadamente 20,7 minutos.
d. aproximadamente 15 minutos.
e. aproximadamente 19,7 minutos.
Passo 1
Primeiro, vamos entender o que a questão está pedindo. Temos um vinho tinto que está na adega a
e é levado para uma sala que está a
. Sabemos que leva 10 minutos para o vinho aquecer até
. A pergunta é: quanto tempo levará para o vinho alcançar
? Para resolver isso, vamos usar a Lei de Resfriamento de Newton, que nos ajuda a entender como a temperatura de um objeto muda ao longo do tempo quando está em contato com um ambiente de temperatura diferente.
Passo 2
A Lei de Resfriamento de Newton diz que a taxa de mudança da temperatura de um objeto é proporcional à diferença entre a temperatura do objeto e a temperatura do ambiente. A fórmula dessa lei é:
Onde:
-
é a temperatura do objeto no tempo . -
é a temperatura do ambiente. -
é a temperatura inicial do objeto. -
é a constante de resfriamento. -
é a base do logaritmo natural.
Passo 3
Vamos identificar os valores que já conhecemos:
- Temperatura inicial do vinho,
- Temperatura do ambiente,
- Tempo para chegar a
, minutos - Temperatura que queremos encontrar,
Passo 4
Primeiro, vamos encontrar a constante
. Usamos a informação de que em 10 minutos o vinho chega a
:
Simplificando:
Agora, tiramos o logaritmo natural de ambos os lados para resolver para
:
Passo 5
Agora que temos
, vamos usar a fórmula da Lei de Resfriamento para encontrar o tempo necessário para o vinho atingir
:
Simplificando:
Tirando o logaritmo natural de ambos os lados:
Substituindo
:
Passo 6
Agora, vamos calcular os valores numéricos:
Então:
Resposta
A temperatura do vinho atingirá
aproximadamente em 19,7 minutos.
Exercícios relacionados
<html><head></head><body><div>Calcule a derivada da função<br>
<span class=“math-inline “>
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<html><head></head><body><div>Uma bateria de 12 volts é conectada a um circuito em série no qual a indutância é de 1/2 henry e a resistência, 10 ohms. Determine a correte i se a corrente inicial é zero. L.di/dt <span class="math-inline ">
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<mjx-container class=“MathJax” jax=“SVG”><svg style=“vertical-align: -0.566ex;” xmlns=“http://www.w3.org/2000/svg” width=“11.252ex” height=“2.262ex” role=“img” focusable=“false” viewBox=“0 -750 4973.6 1000”><title>+ Ri = E(t)</title><g stroke=“currentColor” fill=“currentColor” stroke-width=“0” transform=“scale(1,-1)”><g data-mml-node=“math”><g data-mml-node=“mo”><path data-c=“2B” d=“M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z”></path></g><g data-mml-node=“TeXAtom” data-mjx-texclass=“ORD” transform=“translate(778,0)”><g data-mml-node=“mi”><path data-c=“52” d=“M130 622Q123 629 119 631T103 634T60 637H27V683H202H236H300Q376 683 417 677T500 648Q595 600 609 517Q610 512 610 501Q610 468 594 439T556 392T511 361T472 343L456 338Q459 335 467 332Q497 316 516 298T545 254T559 211T568 155T578 94Q588 46 602 31T640 16H645Q660 16 674 32T692 87Q692 98 696 101T712 105T728 103T732 90Q732 59 716 27T672 -16Q656 -22 630 -22Q481 -16 458 90Q456 101 456 163T449 246Q430 304 373 320L363 322L297 323H231V192L232 61Q238 51 249 49T301 46H334V0H323Q302 3 181 3Q59 3 38 0H27V46H60Q102 47 111 49T130 61V622ZM491 499V509Q491 527 490 539T481 570T462 601T424 623T362 636Q360 636 340 636T304 637H283Q238 637 234 628Q231 624 231 492V360H289Q390 360 434 378T489 456Q491 467 491 499Z”></path><path data-c=“69” d=“M69 609Q69 637 87 653T131 669Q154 667 171 652T188 609Q188 579 171 564T129 549Q104 549 87 564T69 609ZM247 0Q232 3 143 3Q132 3 106 3T56 1L34 0H26V46H42Q70 46 91 49Q100 53 102 60T104 102V205V293Q104 345 102 359T88 378Q74 385 41 385H30V408Q30 431 32 431L42 432Q52 433 70 434T106 436Q123 437 142 438T171 441T182 442H185V62Q190 52 197 50T232 46H255V0H247Z” transform=“translate(736,0)”></path></g></g><g data-mml-node=“mo” transform=“translate(2069.8,0)”><path data-c=“3D” d=“M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z”></path></g><g data-mml-node=“TeXAtom” data-mjx-texclass=“ORD” transform=“translate(3125.6,0)”><g data-mml-node=“mi”><path data-c=“45” d=“M128 619Q121 626 117 628T101 631T58 634H25V680H597V676Q599 670 611 560T625 444V440H585V444Q584 447 582 465Q578 500 570 526T553 571T528 601T498 619T457 629T411 633T353 634Q266 634 251 633T233 622Q233 622 233 621Q232 619 232 497V376H286Q359 378 377 385Q413 401 416 469Q416 471 416 473V493H456V213H416V233Q415 268 408 288T383 317T349 328T297 330Q290 330 286 330H232V196V114Q232 57 237 52Q243 47 289 47H340H391Q428 47 452 50T505 62T552 92T584 146Q594 172 599 200T607 247T612 270V273H652V270Q651 267 632 137T610 3V0H25V46H58Q100 47 109 49T128 61V619Z”></path></g></g><g data-mml-node=“mo” transform=“translate(3806.6,0)”><path data-c=“28” d=“M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z”></path></g><g data-mml-node=“TeXAtom” data-mjx-texclass=“ORD” transform=“translate(4195.6,0)”><g data-mml-node=“mi”><path data-c=“74” d=“M27 422Q80 426 109 478T141 600V615H181V431H316V385H181V241Q182 116 182 100T189 68Q203 29 238 29Q282 29 292 100Q293 108 293 146V181H333V146V134Q333 57 291 17Q264 -10 221 -10Q187 -10 162 2T124 33T105 68T98 100Q97 107 97 248V385H18V422H27Z”></path></g></g><g data-mml-node=“mo” transform=“translate(4584.6,0)”><path data-c=“29” d=“M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z”></path></g></g></g></svg></mjx-container></span>.</div>
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Escreva os quatro primeiros termos das séries seguintes. Determine a convergência e caso sejam convergentes, determine a sua soma.
.
Calcule o limite
.