make sure you show all your work- draw a diagram if necessary, define your variables, write equation(s), solve, check and answer the question.
1) paula mixes 2 cups of sunflower seeds with 3 cups of raisins to make a snack for her hike. she needs the mixture to provide her 2900 calories. find the number of calories/cup in raisins if it is 400 less than the calories/cup in sunflower seeds.
2) a health food store sells a mixture of raisins and roasted nuts. raisins sell for $3.50/kg and nuts sell for $4.75/kg. how many kilograms of each should be mixed to make 20 kg of mixture at $4/kg?
3) a collection of 52 dimes and nickels is worth $4.50. how many nickels are there?
4) a person can row a boat six miles per hour in still water. in a river where the current is two miles per hour, it takes him thirty minutes longer to row a certain distance upstream than to row the same distance downstream. find how long it takes him to row upstream, how long to row downstream, and how many miles he rows.
5) suppose two sisters live 240 miles apart. one sister has three young children who are planning to visit their aunt for a week. to prevent driving so far, the sisters agree to leave at the same time, drive toward each other, and meet somewhere along the route. the sister with the three children tends to drive carefully and obey the speed limit. her average rate of speed is 70 mph. the other sister drives too fast, and her average rate of speed is 80 mph. how long will it take the two sisters to meet each other to transfer the children?
6) one of the legs of a right triangle is seven units longer than the other. the area of the triangle is 6 times the shortest leg. what is the length of the hypotenuse?
7) how many gallons of cream that is 22% fat must be mixed with milk that is 6% fat to produce 30 gallons of cream that is 8% fat?
8) how much water should be mixed with 15 ml of pure alcohol so as to obtain 15% alcohol solution?
9) jenny is making jewelry for an arts and crafts show. she would like to make at least $100 in sales. she estimates that she will sell at most 50 pieces of jewelry. the bracelets that she is selling cost $2 and the necklaces cost $3.
·write a system of inequalities to represent this situation.
·graph the system of inequalities on a graph paper
·give two possible combinations of bracelets and necklaces that can be sold in order for jenny to meet her goal. justify your answer.

Question

Accepted Answer

### 1.
# Explanation:
## Step1: Define variables
Let $x$ be the calories per cup of sunflower seeds and $y$ be the calories per cup of raisins.
## Step2: Set up the equation
We know that $2x + 3y=2900$ and $x=y + 400$. Substitute $x=y + 400$ into $2x+3y = 2900$:
$2(y + 400)+3y=2900$
## Step3: Expand and simplify
$2y+800 + 3y=2900$
$5y=2900 - 800$
$5y=2100$
## Step4: Solve for $y$
$y=\frac{2100}{5}=420$
# Answer:
420 calories per cup of raisins

### 2.
# Explanation:
## Step1: Define variables
Let $x$ be the mass of raisins (in kg) and $y$ be the mass of nuts (in kg).
We have the system of equations: $x + y=20$ (total mass) and $3.5x+4.75y=4	imes20$ (total - cost).
From $x + y=20$, we get $x = 20 - y$.
## Step2: Substitute into the cost - equation
$3.5(20 - y)+4.75y=80$
## Step3: Expand and simplify
$70-3.5y + 4.75y=80$
$1.25y=80 - 70$
$1.25y=10$
## Step4: Solve for $y$
$y=\frac{10}{1.25}=8$
## Step5: Solve for $x$
$x=20 - 8 = 12$
# Answer:
12 kg of raisins and 8 kg of nuts

### 3.
# Explanation:
## Step1: Define variables
Let $d$ be the number of dimes and $n$ be the number of nickels.
We have the system of equations: $d + n=52$ and $0.1d+0.05n = 4.50$.
From $d + n=52$, we get $d=52 - n$.
## Step2: Substitute into the value - equation
$0.1(52 - n)+0.05n=4.50$
## Step3: Expand and simplify
$5.2-0.1n+0.05n=4.50$
$-0.05n=4.50 - 5.2$
$-0.05n=-0.7$
## Step4: Solve for $n$
$n=\frac{-0.7}{-0.05}=14$
# Answer:
14 nickels

### 4.
# Explanation:
## Step1: Define variables
Let $d$ be the distance (in miles) and $t_d$ be the time taken downstream and $t_u$ be the time taken upstream.
The speed downstream is $6 + 2=8$ mph and the speed upstream is $6-2 = 4$ mph.
We know that $t_u=t_d+\frac{30}{60}$ (since 30 minutes = 0.5 hours), and using $t=\frac{d}{s}$ (where $t$ is time, $d$ is distance, and $s$ is speed), $\frac{d}{4}=\frac{d}{8}+0.5$
## Step2: Solve the equation
Multiply through by 8 to clear the fractions: $2d=d + 4$
$2d-d=4$
$d = 4$ miles
$t_u=\frac{d}{4}=\frac{4}{4}=1$ hour
$t_d=\frac{d}{8}=\frac{4}{8}=0.5$ hour
# Answer:
The distance is 4 miles, time upstream is 1 hour, time downstream is 0.5 hour

### 5.
# Explanation:
## Step1: Use the formula $d=st$
Let $t$ be the time they drive until they meet.
The sum of the distances they travel is 240 miles. The distance one sister travels is $d_1 = 70t$ and the other is $d_2=80t$.
So, $70t+80t=240$
## Step2: Combine like - terms and solve for $t$
$150t=240$
$t=\frac{240}{150}=1.6$ hours
# Answer:
1.6 hours

### 6.
# Explanation:
## Step1: Define variables
Let the shorter leg of the right - triangle be $x$ units. Then the other leg is $x + 7$ units.
The area of a right - triangle is $A=\frac{1}{2}ab$. Here, $A = 6x$. So, $\frac{1}{2}x(x + 7)=6x$
## Step2: Solve the area equation
Multiply both sides by 2 to get $x(x + 7)=12x$
$x^2+7x=12x$
$x^2+7x-12x=0$
$x^2 - 5x=0$
$x(x - 5)=0$
$x = 5$ (we ignore $x = 0$).
The legs are $x = 5$ and $x + 7=12$
## Step3: Use the Pythagorean theorem
$c=\sqrt{5^2+12^2}=\sqrt{25 + 144}=\sqrt{169}=13$
# Answer:
13 units

### 7.
# Explanation:
## Step1: Define variables
Let $x$ be the volume of 8% fat cream and $y$ be the volume of 6% fat milk.
We have $x + y=30$ and $0.08x+0.06y=0.22	imes30$
From $x + y=30$, we get $y = 30 - x$
## Step2: Substitute into the fat - content equation
$0.08x+0.06(30 - x)=6.6$
## Step3: Expand and simplify
$0.08x+1.8-0.06x=6.6$
$0.02x=6.6 - 1.8$
$0.02x=4.8$
$x = 240$
$y=30 - 240=-210$ (This is wrong. Let's correct the setup. Let $x$ be the volume of 22% fat cream and $y$ be the volume of 6% fat milk)
$x + y=30$ and $0.22x+0.06y=0.08	imes30$
From $x + y=30$, $y = 30 - x$
$0.22x+0.06(30 - x)=2.4$
$0.22x + 1.8-0.06x=2.4$
$0.16x=2.4 - 1.8$
$0.16x=0.6$
$x=\frac{0.6}{0.16}=\frac{60}{16}=\frac{15}{4}=3.75$ gallons
$y=30 - 3.75 = 26.25$ gallons
# Answer:
3.75 gallons of 22% fat cream and 26.25 gallons of 6% fat milk

### 8.
# Explanation:
## Step1: Define variables
Let $x$ be the volume of water (in mL).
The amount of alcohol is 15 mL. The total volume of the solution is $x + 15$ mL.
We want the concentration of alcohol to be 15%, so $\frac{15}{x + 15}=0.15$
## Step2: Solve the equation
$15=0.15(x + 15)$
$15=0.15x+2.25$
$0.15x=15 - 2.25$
$0.15x=12.75$
$x=\frac{12.75}{0.15}=85$ mL
# Answer:
85 mL of water

### 9.
# Explanation:
## Step1: Write the inequalities
Let $b$ be the number of bracelets and $n$ be the number of necklaces.
The number - of - items inequality is $b + n\leq50$
The money - making inequality is $2b+3n\geq100$
## Step2: Graph the inequalities
For $b + n\leq50$, when $b = 0$, $n = 50$ and when $n = 0$, $b = 50$. The boundary line is $b + n=50$ and we shade the region below the line (including the line) since it is $\leq$.
For $2b+3n\geq100$, when $b = 0$, $n=\frac{100}{3}\approx33.33$ and when $n = 0$, $b = 50$. The boundary line is $2b + 3n=100$ and we shade the region above the line (including the line) since it is $\geq$.
## Step3: Find possible combinations
We can find the intersection points of the boundary lines and then pick integer points in the feasible region.
If $b + n=50$ and $2b+3n=100$, from $b = 50 - n$, $2(50 - n)+3n=100$, $100-2n+3n=100$, $n = 0$, $b = 50$
Another point: Let $n = 10$, then from $b + n=50$, $b = 40$, and $2b+3n=2	imes40+3	imes10=80 + 30=110\geq100$
Two possible combinations are $(50,0)$ (50 bracelets and 0 necklaces) and $(40,10)$ (40 bracelets and 10 necklaces)
# Answer:
The system of inequalities is $\begin{cases}b + n\leq50\2b+3n\geq100\end{cases}$
Two possible combinations are (50,0) and (40,10)

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QUESTION IMAGE

Question

Explanation:

1.

Step1: Define variables

Step2: Set up the equation

Step3: Expand and simplify

Step4: Solve for $y$

Step1: Define variables

Step2: Substitute into the cost - equation

Step3: Expand and simplify

Step4: Solve for $y$

Step5: Solve for $x$

Step1: Define variables

Step2: Substitute into the value - equation

Step3: Expand and simplify

Step4: Solve for $n$

Answer:

2.