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Solve linear equations with variables on both sides

Each cup of cooked brown rice has 216216 calories and 55 grams of protein. Each cup of kidney beans has 220220 calories and 1616 grams of protein. Which of the following systems of equations could be used to determine the amount of rice, r r , in cups, and the amount of beans, b b , in cups, that should be eaten in order to consume 435435 calories and 1818.2525 grams of protein?\newlineChoose 11 answer:\newline(A) 216r+5b=435 216 r+5 b=435 \newline220r+16b=18.25 220 r+16 b=18.25 \newline(B) 216b+5r=435 216 b+5 r=435 \newline220b+16r=18.25 220 b+16 r=18.25 \newline(C) 216r+220b=435 216 r+220 b=435 \newline5r+16b=18.25 5 r+16 b=18.25 \newline(D) 216b+220r=435 216 b+220 r=435 \newline5b+16r=18.25 5 b+16 r=18.25
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Lara tried to solve the differential equation dydx=xy+2x \frac{d y}{d x}=x y+2 x . This is her work:\newlinedydx=xy+2x \frac{d y}{d x}=x y+2 x \newlineStep 11: dydx=x(y+2)\quad \frac{d y}{d x}=x(y+2) \newlineStep 22: ∫(y+2)−1dy=∫xdx \quad \int(y+2)^{-1} d y=\int x d x \newlineStep 33: ln⁡∣y+2∣=x22+C1 \quad \ln |y+2|=\frac{x^{2}}{2}+C_{1} \newlineStep 44: eln⁡∣y+2∣=ex22+C1 \quad e^{\ln |y+2|}=e^{\frac{x^{2}}{2}+C_{1}} \newlineStep 55: ∣y+2∣=ex22eC1 \quad|y+2|=e^{\frac{x^{2}}{2}} e^{C_{1}} \newlineStep 66: y+2=Cex22 \quad y+2=C e^{\frac{x^{2}}{2}} \newlineStep 77: y=Cex22−2\quad y=C e^{\frac{x^{2}}{2}}-2 \newlineIs Lara's work correct? If not, what is her mistake?\newlineChoose 11 answer:\newline(A) Lara's work is correct.\newline(B) Step 11 is incorrect. We're not allowed to factor an expression when solving differential equations.\newline(C) Step 2 \mathbf{2} is incorrect. The separation of variables wasn't done correctly.\newline(D) Step 44 is incorrect. The right-hand side of the equation should be ez22+C1 e^{\frac{z^{2}}{2}}+C_{1} .
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Aiden was given this problem:\newlineThe radius r(t) r(t) of the base of a cylinder is increasing at a rate of 11 meter per hour and the height h(t) h(t) of the cylinder is decreasing at a rate of 44 meters per hour. At a certain instant t0 t_{0} , the base radius is 55 meters and the height is 88 meters. What is the rate of change of the volume V(t) V(t) of the cylinder at that instant?\newlineWhich equation should Aiden use to solve the problem?\newlineChoose 11 answer:\newline(A) V(t)=π[r(t)]2+2π⋅r(t)⋅h(t) V(t)=\pi[r(t)]^{2}+2 \pi \cdot r(t) \cdot h(t) \newline(B) V(t)=π[r(t)]2⋅h(t) V(t)=\pi[r(t)]^{2} \cdot h(t) \newline(C) V(t)=π[r(t)]2+π⋅r(t)[r(t)]2+[h(t)]2 V(t)=\pi[r(t)]^{2}+\pi \cdot r(t) \sqrt{[r(t)]^{2}+[h(t)]^{2}} \newline(D) V(t)=π[r(t)]2⋅h(t)3 V(t)=\frac{\pi[r(t)]^{2} \cdot h(t)}{3}
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