67. nutrition one - eight - ounce glass of apple juice and one eight - ounce glass of orange juice contain a…

67. nutrition one - eight - ounce glass of apple juice and one eight - ounce glass of orange juice contain a total of 227 milligrams of vitamin c. two eight - ounce glasses of apple juice and three eight - ounce glasses of orange juice contain a total of 578 milligrams of vitamin c. how much vitamin c is in an eight - ounce glass of each type of juice?\n68. airplane speed two planes start from los angeles international airport and fly in opposite directions. the second plane starts \\(\\frac{1}{2}\\) hour after the first plane, but its speed is 80 kilometers per hour faster. two hours after the first plane departs, the planes are 3200 kilometers apart. find the airspeed of each plane.

67. nutrition one - eight - ounce glass of apple juice and one eight - ounce glass of orange juice contain a total of 227 milligrams of vitamin c. two eight - ounce glasses of apple juice and three eight - ounce glasses of orange juice contain a total of 578 milligrams of vitamin c. how much vitamin c is in an eight - ounce glass of each type of juice?\n68. airplane speed two planes start from los angeles international airport and fly in opposite directions. the second plane starts \\(\\frac{1}{2}\\) hour after the first plane, but its speed is 80 kilometers per hour faster. two hours after the first plane departs, the planes are 3200 kilometers apart. find the airspeed of each plane.

Answer

Explanation:

Step1: Set up equations for vitamin - C problem

Let $x$ be the amount of vitamin C in an eight - ounce glass of apple juice and $y$ be the amount of vitamin C in an eight - ounce glass of orange juice. We have the system of equations: $x + y=227$ (Equation 1: One glass of each has 227 mg of vitamin C) $2x + 3y=578$ (Equation 2: Two glasses of apple juice and three glasses of orange juice have 578 mg of vitamin C) From Equation 1, we can express $x = 227 - y$.

Step2: Substitute $x$ into Equation 2

Substitute $x = 227 - y$ into $2x+3y = 578$. $2(227 - y)+3y=578$. Expand the left - hand side: $454-2y + 3y=578$. Combine like terms: $454 + y=578$. Subtract 454 from both sides: $y=578 - 454=124$.

Step3: Find the value of $x$

Substitute $y = 124$ into $x = 227 - y$. $x=227-124 = 103$.

Step4: Set up equation for plane speed problem

Let the speed of the first plane be $v$ km/h. Then the speed of the second plane is $v + 80$ km/h. The first plane flies for $t_1=2$ hours and the second plane flies for $t_2=2-\frac{1}{2}=\frac{3}{2}$ hours. The distance formula is $d = vt$. The sum of their distances is 3200 km. So $2v+\frac{3}{2}(v + 80)=3200$. Expand the left - hand side: $2v+\frac{3}{2}v+120 = 3200$. Combine like terms: $\left(2+\frac{3}{2}\right)v=3200 - 120$. $\frac{4v+3v}{2}=3080$. $\frac{7v}{2}=3080$. Multiply both sides by $\frac{2}{7}$: $v=\frac{3080\times2}{7}=880$ km/h. The speed of the second plane is $v + 80=880+80 = 960$ km/h.

Answer:

The amount of vitamin C in an eight - ounce glass of apple juice is 103 milligrams. The speed of the first plane is 880 km/h and the speed of the second plane is 960 km/h.