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3 years ago

Sally is having a party and she has invited nine of her friends.

Mathematics

1 answer:

Alexeev081 [22]3 years ago

3 0

Answer:

1) 5 limes, 5 lemons, 1,000 mL of pineapple juice, 2,000 mL of lemonade, 1,000 of orange juice

2) Limes = $2, Lemons = $2.5, Lemonade = $1.60, Orange Juice = 80c, Pineapple Juice = 90c

Step-by-step explanation:

1/2 limes · 10 = 5

1/2 lemons · 10 = 5

100 mL of pineapple juice · 10 = 1,000 mL

200 mL of lemonade · 10 = 2,000 mL

100 mL of orange juice · 10 = 1,000 mL

Cost:

5 limes · 0.4 = $2

5 lemons · 0.5 = $2.5

mL to L

1,000 mL of pineapple juice = 1 L, this costs 90¢

2,000 mL of lemonade = 2 L, this costs $1.60

1,000 mL of orange juice = 1 L, this costs 80¢

Overall:

5 limes for $2

5 lemons for $2.5

1 L of of pineapple juice for 90¢

2 L of lemonade for $1.60

1 L of orange juice for 80¢

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In a previous exercise we formulated a model for learning in the form of the differential equation dp dt = k(m − p) where p(t) m

DaniilM [7]

I assume you mean

$\dfrac{dP}{dt} = k(M-P)$

ANSWER
An expression for P(t) is

$P = M - Me^{-kt}$

EXPLANATION
This is a separable differential equation. Treat M and k as constants. Then we can divide both sides by M - P to get the P term with the differential dP and multiply both sides by dt to separate dt from the P terms

$\begin{aligned} \dfrac{dP}{dt} &= k(M-P) \\ \dfrac{dP}{M-P} &= k\, dt
\end{aligned}$

Integrate both sides of the equation.

$\begin{aligned}
\int \dfrac{dP}{M-P} &= \int k\, dt \\
-\ln|M-P| &= kt + C \\
\ln|M-P| &= -kt - C\end{aligned}$

Note that for the left-hand side, u-substitution gives us

$u = M - P \implies du = -1dP \implies dP = -du$

hence why $\int \frac{dP}{M-P} \ne \ln|M - P|$

Now we use the definition of the logarithm to convert into exponential form.

The definition is

$\ln(a) = b \iff \log_e(a) = b \iff e^b = a$

so applying it here, we get

$\begin{aligned} \ln|M-P| &= -kt - C \\ |M - P| &= e^{-kt - C} \\ 
M - P &= \pm e^{-kt - C} 
 \end{aligned}$

Exponent properties can be used to address the constant C. We use $x^{a} \cdot x^{b} = x^{a+b}$ here:

$\begin{aligned}
 M - P &= \pm e^{-kt - C} \\
M - P &= \pm e^{- C - kt} \\ 
M - P &= \pm e^{- C + (- kt)} \\ 
M - P &= \pm e^{- C} \cdot e^{- kt} \\ 
M - P &= Ke^{- kt} && (\text{\footnotesize Let $K = \pm e^{-kt}$ }) \\ 
M &= Ke^{- kt} + P\\
P &= M - Ke^{- kt}
\end{aligned}$

If we assume that P(0) = 0, then set t = 0 and P = 0

$\begin{aligned} 
0 &= M - Ke^{- k\cdot 0} \\
0 &= M - K \cdot 1 \\
M &= K
 \end{aligned}
$

Substituting into our original equation, we get our final answer of

$P = M - Me^{-kt}$

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Answer:(3-8

Step-by-step explanation:

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2 years ago

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Yuliya22 [10]

Answer:

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Step-by-step explanation:

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brilliants [131]

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Angelina_Jolie [31]

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