Math / Geometry

QuestionThe triangular region shows the number of possible ounces of platinum, $x$ , and the number of possible ounces of silver, $y$ , a jewelry maker can mix together to make a certain piece of jewelry.
Which combination of platinum and silver can the jewelry maker use? 16 oz of platinum and 30 oz of silver 25 oz of platinum and 20 oz of silver 10 oz of platinum and 10 oz of silver 12 oz of platinum and 20 oz of silver

Studdy Solution

STEP 1

1. The triangular region represents the feasible combinations of platinum ( $x$ ) and silver ( $y$ ) the jewelry maker can use.
2. The vertices of the triangular region are approximately at $(0, 20)$ , $(20, 0)$ , and $(10, 10)$ .
3. The problem requires checking whether the given combinations of platinum and silver lie within this triangular region.

STEP 2

1. Define the equations of the lines forming the boundaries of the triangular region.
2. Determine if each combination of platinum and silver lies within the triangular region by checking if they satisfy the inequalities formed by these boundary lines.

STEP 3

Define the line passing through the points $(0, 20)$ and $(20, 0)$ .
The slope $m$ of the line is: $m = \frac{0 - 20}{20 - 0} = -1$
The equation of the line in slope-intercept form $y = mx + b$ is: $y = -x + 20$

STEP 4

Define the line passing through the points $(0, 20)$ and $(10, 10)$ .
The slope $m$ of the line is: $m = \frac{10 - 20}{10 - 0} = -1$
The equation of the line in slope-intercept form $y = mx + b$ is: $y = -x + 20$ (Note: This line coincides with the previous one, so it does not provide a new boundary.)

STEP 5

Define the line passing through the points $(20, 0)$ and $(10, 10)$ .
The slope $m$ of the line is: $m = \frac{10 - 0}{10 - 20} = -1$
The equation of the line in slope-intercept form $y = mx + b$ is: $y = -x + 20$ (Note: This line also coincides with the previous one.)

STEP 6

Define the line passing through the points $(10, 10)$ and $(0, 20)$ (new boundary).
The slope $m$ of the line is: $m = \frac{20 - 10}{0 - 10} = -1$
The equation of the line in slope-intercept form $y = mx + b$ is: $y = -x + 20$

STEP 7

Define the line passing through the points $(20, 0)$ and $(10, 10)$ (another new boundary).
The slope $m$ of the line is: $m = \frac{10 - 0}{10 - 20} = -1$
The equation of the line in slope-intercept form $y = mx + b$ is: $y = -x + 10$

STEP 8

Determine if the point $(16, 30)$ lies within the triangular region by checking if it satisfies the boundary equations.
For $y = -x + 20$ : $30 \leq -16 + 20 \implies 30 \leq 4 \text{ (False)}$
Since one inequality is not satisfied, the point $(16, 30)$ is outside the triangular region.

STEP 9

Determine if the point $(25, 20)$ lies within the triangular region by checking if it satisfies the boundary equations.
For $y = -x + 20$ : $20 \leq -25 + 20 \implies 20 \leq -5 \text{ (False)}$
Since one inequality is not satisfied, the point $(25, 20)$ is outside the triangular region.

STEP 10

Determine if the point $(10, 10)$ lies within the triangular region by checking if it satisfies the boundary equations.
For $y = -x + 20$ : $10 \leq -10 + 20 \implies 10 \leq 10 \text{ (True)}$
For $y = -x + 10$ : $10 \leq -10 + 10 \implies 10 \leq 0 \text{ (False)}$
Since one inequality is not satisfied, the point $(10, 10)$ is outside the triangular region.

STEP 11

Determine if the point $(12, 20)$ lies within the triangular region by checking if it satisfies the boundary equations.
For $y = -x + 20$ : $20 \leq -12 + 20 \implies 20 \leq 8 \text{ (False)}$
Since one inequality is not satisfied, the point $(12, 20)$ is outside the triangular region.

STEP 12

Determine if the point $(10, 10)$ lies within the triangular region by checking if it satisfies the boundary equations.
For $y = -x + 20$ : $10 \leq -10 + 20 \implies 10 \leq 10 \text{ (True)}$
For $y = -x + 10$ : $10 \leq -10 + 10 \implies 10 \leq 0 \text{ (False)}$
Since one inequality is not satisfied, the point $(10, 10)$ is outside the triangular region.
Solution: None of the given combinations can be used by the jewelry maker.

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Studdy Solution

STEP 1

STEP 2

1. Define the equations of the lines forming the boundaries of the triangular region.
2. Determine if each combination of platinum and silver lies within the triangular region by checking if they satisfy the inequalities formed by these boundary lines.

STEP 3

Define the line passing through the points $(0, 20)$ and $(20, 0)$ .
The slope $m$ of the line is: $m = \frac{0 - 20}{20 - 0} = -1$
The equation of the line in slope-intercept form $y = mx + b$ is: $y = -x + 20$

STEP 4

STEP 5

Define the line passing through the points $(20, 0)$ and $(10, 10)$ .
The slope $m$ of the line is: $m = \frac{10 - 0}{10 - 20} = -1$
The equation of the line in slope-intercept form $y = mx + b$ is: $y = -x + 20$ (Note: This line also coincides with the previous one.)

STEP 6

Define the line passing through the points $(10, 10)$ and $(0, 20)$ (new boundary).
The slope $m$ of the line is: $m = \frac{20 - 10}{0 - 10} = -1$
The equation of the line in slope-intercept form $y = mx + b$ is: $y = -x + 20$

STEP 7

Define the line passing through the points $(20, 0)$ and $(10, 10)$ (another new boundary).
The slope $m$ of the line is: $m = \frac{10 - 0}{10 - 20} = -1$
The equation of the line in slope-intercept form $y = mx + b$ is: $y = -x + 10$

STEP 8

Determine if the point $(16, 30)$ lies within the triangular region by checking if it satisfies the boundary equations.
For $y = -x + 20$ : $30 \leq -16 + 20 \implies 30 \leq 4 \text{ (False)}$
Since one inequality is not satisfied, the point $(16, 30)$ is outside the triangular region.

STEP 9

Determine if the point $(25, 20)$ lies within the triangular region by checking if it satisfies the boundary equations.
For $y = -x + 20$ : $20 \leq -25 + 20 \implies 20 \leq -5 \text{ (False)}$
Since one inequality is not satisfied, the point $(25, 20)$ is outside the triangular region.

STEP 10

STEP 11

Determine if the point $(12, 20)$ lies within the triangular region by checking if it satisfies the boundary equations.
For $y = -x + 20$ : $20 \leq -12 + 20 \implies 20 \leq 8 \text{ (False)}$
Since one inequality is not satisfied, the point $(12, 20)$ is outside the triangular region.

STEP 12

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Studdy Solution

STEP 1

STEP 2

1. Define the equations of the lines forming the boundaries of the triangular region.2. Determine if each combination of platinum and silver lies within the triangular region by checking if they satisfy the inequalities formed by these boundary lines.

STEP 3

Define the line passing through the points (0,20)(0, 20)(0,20) and (20,0)(20, 0)(20,0).The slope mmm of the line is: m=0−2020−0=−1 m = \frac{0 - 20}{20 - 0} = -1 m=20−00−20​=−1The equation of the line in slope-intercept form y=mx+by = mx + by=mx+b is: y=−x+20 y = -x + 20 y=−x+20

STEP 4

STEP 5

STEP 6

STEP 7

STEP 8

STEP 9

STEP 10

STEP 11

STEP 12

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Define the equations of the lines forming the boundaries of the triangular region.
2. Determine if each combination of platinum and silver lies within the triangular region by checking if they satisfy the inequalities formed by these boundary lines.

Define the line passing through the points $(0, 20)$ and $(20, 0)$ .
The slope $m$ of the line is: $m = \frac{0 - 20}{20 - 0} = -1$
The equation of the line in slope-intercept form $y = mx + b$ is: $y = -x + 20$