1.) A cooler contains fourteen bottles of Sports drinks: Seven lemon-lime flavored and seven orange flavored. You randomly grab a bottle and give it to your friend. Then, you randomly grab a bottle for yourself. What is the probability that you and your friend both get lemon-lime?
2.) Find the distance between (3,4) and (-5,8)
3.) Points A, B, and C are collinear. Point B is between A and C. Find X if AB = 10x + 1, BC = 5x + 1, and AC = 17.
4.) Two sides of a triangle have the following measures: 15 and 39. What is the range of possible measures for the third side?
5.) Line segment AB has endpoints A(10,4) and B(2,8). Find the coordinates of the point that divides the line segment directed from A to B in the ratio of 1:4.
6.) A softball player bats twice in a game. Each at bat results in an out, getting on base, or hitting a home run. what is the number of possible outcomes in the sample space?
7.) You flip a coin and then roll a fair six-sided number cube. what is the probability the coin lands heads up and the number cube shows a number 3?
8.) There are twelve shirts in your closet. Five blue, four green, and three red. you randomly select one to wear. what is the probability it is green or blue?
1.) The probability that you grab a lemon-lime flavored bottle is 7/14, or 1/2, because there are seven lemon-lime bottles out of a total of fourteen. After giving that bottle to your friend, the probability that your friend also gets a lemon-lime bottle is now 6/13, because there are now six lemon-lime bottles left out of a total of thirteen. Therefore, the probability that you and your friend both get lemon-lime is (1/2) * (6/13) = 3/13.
2.) The distance between two points (x1, y1) and (x2, y2) in a coordinate plane can be found using the distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2). In this case, the coordinates are (3,4) and (-5,8). Plugging these values into the formula, we get d = sqrt((-5 - 3)^2 + (8 - 4)^2) = sqrt((-8)^2 + (4)^2) = sqrt(64 + 16) = sqrt(80) ≈ 8.94.
3.) Since points A, B, and C are collinear and B is between A and C, the sum of the lengths of AB and BC should be equal to the length of AC. Therefore, we have the equation AB + BC = AC. Substituting the given values, we get (10x + 1) + (5x + 1) = 17. Combining like terms, we have 15x + 2 = 17. Subtracting 2 from both sides, we get 15x = 15. Finally, dividing both sides by 15, we find x = 1.
4.) For any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Therefore, for a triangle with sides measuring 15 and 39, the third side must be greater than the difference between the lengths of the other two sides but less than the sum of the lengths of the other two sides. So, the range of possible measures for the third side is 39 - 15 < third side < 39 + 15, which simplifies to 24 < third side < 54.
5.) To find the coordinates of the point that divides the line segment AB in the ratio of 1:4, we can use the section formula. The section formula states that the coordinates of the point P(x, y) that divides the line segment with endpoints A(x1, y1) and B(x2, y2) in the ratio m:n are given by the formulas x = (nx1 + mx2)/(m + n) and y = (ny1 + my2)/(m + n). In this case, the coordinates of A are (10, 4), the coordinates of B are (2, 8), and the ratio is 1:4. Plugging these values into the formulas, we find x = (4*10 + 1*2)/(1 + 4) = 42/5 and y = (4*4 + 1*8)/(1 + 4) = 24/5. Therefore, the coordinates of the point that divides the line segment AB in the ratio of 1:4 are (42/5, 24/5).
6.) For each at bat, there are three possible outcomes - an out, getting on base, or hitting a home run. Since the player bats twice in a game, the number of possible outcomes in the sample space is 3 * 3 = 9.
7.) The probability of the coin landing heads up is 1/2, since there are two equally likely outcomes, heads and tails. The probability of the number cube showing a number 3 is 1/6, since there are six equally likely outcomes, numbers 1 to 6. The probability of both events happening simultaneously (coin landing heads up and number cube showing a number 3) is the product of their individual probabilities, which is (1/2) * (1/6) = 1/12.
8.) There are a total of 12 shirts in the closet, with 5 blue shirts, 4 green shirts, and 3 red shirts. Therefore, the probability of randomly selecting a green or blue shirt is (5 + 4)/12 = 9/12 = 3/4.
1.) To find the probability that you and your friend both get lemon-lime, we need to consider the total number of possible outcomes and the number of favorable outcomes.
Total number of bottles: 14 (since there are 14 bottles in the cooler)
Number of lemon-lime bottles: 7 (half of the total bottles)
Number of favorable outcomes: 7 (since both you and your friend need to get lemon-lime)
The probability can be calculated by dividing the number of favorable outcomes by the total number of outcomes:
Probability = Number of favorable outcomes / Total number of outcomes
Probability = 7 / 14
Simplifying the fraction, we get:
Probability = 1/2
So, the probability that you and your friend both get lemon-lime is 1/2 or 50%.
2.) To find the distance between two points (x₁, y₁) and (x₂, y₂) in a coordinate plane, we can use the distance formula:
Distance = √((x₂ - x₁)² + (y₂ - y₁)²)
Plugging in the coordinates (3, 4) and (-5, 8) into the formula:
Distance = √((-5 - 3)² + (8 - 4)²)
= √((-8)² + (4)²)
= √(64 + 16)
= √(80)
≈ 8.94
So, the distance between (3, 4) and (-5, 8) is approximately 8.94 units.
3.) Since points A, B, and C are collinear and B is between A and C, we can form a linear equation using the given information.
From the given lengths of AB, BC, and AC, we have the equation:
AB + BC = AC
Substituting the given values:
(10x + 1) + (5x + 1) = 17
Simplifying the equation:
10x + 1 + 5x + 1 = 17
15x + 2 = 17
15x = 17 - 2
15x = 15
x = 15/15
x = 1
So, the value of x is 1.
4.) To find the range of possible measures for the third side of a triangle, we consider the triangle inequality theorem. According to this theorem, for any triangle with sides a, b, and c:
a + b > c
a + c > b
b + c > a
Given that two sides of the triangle have measures 15 and 39, we can substitute these values into the triangle inequality:
15 + 39 > c
c < 54
So, the third side must be less than 54 units in order for it to form a valid triangle.
5.) To find the coordinates of the point that divides the line segment AB in the ratio of 1:4, we can use the section formula. Let's assume the coordinates of the point are (x, y).
The coordinates of point A are (10, 4), and the coordinates of point B are (2, 8).
Using the section formula, we have:
x = (4/5) * 2 + (1/5) * 10 = 8/5 + 10/5 = 18/5 = 3.6
y = (4/5) * 8 + (1/5) * 4 = 32/5 + 4/5 = 36/5 = 7.2
So, the coordinates of the point that divides the line segment AB in the ratio of 1:4 are approximately (3.6, 7.2).
6.) The softball player can have three possible outcomes in each at-bat: out, getting on base, or hitting a home run. Therefore, the number of possible outcomes for each at-bat is 3.
Since the player bats twice, we multiply the number of outcomes for each at-bat:
Number of possible outcomes in the sample space = 3 * 3 = 9
So, there are 9 possible outcomes in the sample space.
7.) The probability of the coin landing heads up is 1/2 since there are two equally likely outcomes - heads or tails.
The probability of the number cube showing a 3 is 1/6 since there is only one favorable outcome out of six possible outcomes (the numbers 1, 2, 3, 4, 5, and 6).
To find the probability of both events occurring, we multiply their individual probabilities:
Probability = (Probability of coin landing heads) * (Probability of number cube showing a 3)
= (1/2) * (1/6)
= 1/12
So, the probability that the coin lands heads up and the number cube shows a number 3 is 1/12.
8.) The total number of shirts in the closet is 12.
The number of blue shirts is 5, and the number of green shirts is 4. So, the number of shirts that are either green or blue is 5 + 4 = 9.
To find the probability of selecting a green or blue shirt, we divide the number of favorable outcomes (9) by the total number of outcomes (12):
Probability = Number of favorable outcomes / Total number of outcomes
= 9 / 12
= 3 / 4
So, the probability of randomly selecting a green or blue shirt from the closet is 3/4 or 75%.