I am having some trouble with my Algrbra 2 class. We have this thing called challenge of the week. Every week, our teacher gives us some problems, but they are problems involving something we haven't been taught yet: quadratic equations. I would very much appreciate any help that you provide! Here are the questions:
1. If V= 12R/(r+R) make R the subject of the equation.
2a. Jane, Maria and Ben each have a collection of marbles. Jane has 15 more marbles than Ben, and Maria has 2 times as many marbles as Ben. All together, they have 95 marbles. Find how many marbles Maria has.
2b. Dave sold 40 tickets for a concert. He sold x tickets at $2 each and y tickets at $3 each. He collected $88. Write down two equations connecting x and y. Solve these two equations to find how many of each kind of ticket he sold.
2c. A rectangle has length (x+5) cm and width of (x-2) cm. Its area is 60 cm^2. Write a quadratic equation, and solve it to fins the length and the width of this rectangle.
Thank you so much! 😄😄😄
1. If V= 12R/(r+R) make R the subject of the equation.
Solve for R:
V = (12 R)/(r+R)
V = (12 R)/(r+R) is equivalent to (12 R)/(r+R) = V:
(12 R)/(r+R) = V
Multiply both sides by r+R:
12 R = V (r+R)
Expand out terms of the right hand side:
12 R = r V+R V
Subtract R V from both sides:
R (12-V) = r V
Divide both sides by 12-V:
Answer: |R = (r V)/(12-V)
2a. Jane, Maria and Ben each have a collection of marbles. Jane has 15 more marbles than Ben, and Maria has 2 times as many marbles as Ben. All together, they have 95 marbles. Find how many marbles Maria has.
Let Ben's marbles =B
Jane has =B+15 marbles
Maria has =2B marbles
B + B + 15 + 2B =95
4B=95 - 15
4B =80 divide both sides by 4
B=20 Ben's marbles
20+15 =35 Jane's marbles
2 x 20 =40 Maria's marbles.
2b. Dave sold 40 tickets for a concert. He sold x tickets at $2 each and y tickets at $3 each. He collected $88. Write down two equations connecting x and y. Solve these two equations to find how many of each kind of ticket he sold
x + y =40
2x + 3y =88
Solve the following system:
{x+y = 40 | (equation 1)
2 x+3 y = 88 | (equation 2)
Swap equation 1 with equation 2:
{2 x+3 y = 88 | (equation 1)
x+y = 40 | (equation 2)
Subtract 1/2 × (equation 1) from equation 2:
{2 x+3 y = 88 | (equation 1)
0 x-y/2 = -4 | (equation 2)
Multiply equation 2 by -2:
{2 x+3 y = 88 | (equation 1)
0 x+y = 8 | (equation 2)
Subtract 3 × (equation 2) from equation 1:
{2 x+0 y = 64 | (equation 1)
0 x+y = 8 | (equation 2)
Divide equation 1 by 2:
{x+0 y = 32 | (equation 1)
0 x+y = 8 | (equation 2)
Collect results:
Answer: |x = 32 and y = 8
2c. A rectangle has length (x+5) cm and width of (x-2) cm. Its area is 60 cm^2. Write a quadratic equation, and solve it to fins the length and the width of this rectangle.
Solve for x:
(x-2) (x+5) = 60
Expand out terms of the left hand side:
x^2+3 x-10 = 60
Add 10 to both sides:
x^2+3 x = 70
Add 9/4 to both sides:
x^2+3 x+9/4 = 289/4
Write the left hand side as a square:
(x+3/2)^2 = 289/4
Take the square root of both sides:
x+3/2 = 17/2 or x+3/2 = -17/2
Subtract 3/2 from both sides:
x = 7 or x+3/2 = -17/2
Subtract 3/2 from both sides:
Answer: |x = 7
1. If V= 12R/(r+R) make R the subject of the equation.
Solve for R:
V = (12 R)/(r+R)
V = (12 R)/(r+R) is equivalent to (12 R)/(r+R) = V:
(12 R)/(r+R) = V
Multiply both sides by r+R:
12 R = V (r+R)
Expand out terms of the right hand side:
12 R = r V+R V
Subtract R V from both sides:
R (12-V) = r V
Divide both sides by 12-V:
Answer: |R = (r V)/(12-V)
2a. Jane, Maria and Ben each have a collection of marbles. Jane has 15 more marbles than Ben, and Maria has 2 times as many marbles as Ben. All together, they have 95 marbles. Find how many marbles Maria has.
Let Ben's marbles =B
Jane has =B+15 marbles
Maria has =2B marbles
B + B + 15 + 2B =95
4B=95 - 15
4B =80 divide both sides by 4
B=20 Ben's marbles
20+15 =35 Jane's marbles
2 x 20 =40 Maria's marbles.
2b. Dave sold 40 tickets for a concert. He sold x tickets at $2 each and y tickets at $3 each. He collected $88. Write down two equations connecting x and y. Solve these two equations to find how many of each kind of ticket he sold
x + y =40
2x + 3y =88
Solve the following system:
{x+y = 40 | (equation 1)
2 x+3 y = 88 | (equation 2)
Swap equation 1 with equation 2:
{2 x+3 y = 88 | (equation 1)
x+y = 40 | (equation 2)
Subtract 1/2 × (equation 1) from equation 2:
{2 x+3 y = 88 | (equation 1)
0 x-y/2 = -4 | (equation 2)
Multiply equation 2 by -2:
{2 x+3 y = 88 | (equation 1)
0 x+y = 8 | (equation 2)
Subtract 3 × (equation 2) from equation 1:
{2 x+0 y = 64 | (equation 1)
0 x+y = 8 | (equation 2)
Divide equation 1 by 2:
{x+0 y = 32 | (equation 1)
0 x+y = 8 | (equation 2)
Collect results:
Answer: |x = 32 and y = 8
2c. A rectangle has length (x+5) cm and width of (x-2) cm. Its area is 60 cm^2. Write a quadratic equation, and solve it to fins the length and the width of this rectangle.
Solve for x:
(x-2) (x+5) = 60
Expand out terms of the left hand side:
x^2+3 x-10 = 60
Add 10 to both sides:
x^2+3 x = 70
Add 9/4 to both sides:
x^2+3 x+9/4 = 289/4
Write the left hand side as a square:
(x+3/2)^2 = 289/4
Take the square root of both sides:
x+3/2 = 17/2 or x+3/2 = -17/2
Subtract 3/2 from both sides:
x = 7 or x+3/2 = -17/2
Subtract 3/2 from both sides:
Answer: |x = 7
