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Elza [17]
3 years ago
15

The sides of a right triangle have a relationship as depicted below, where x is the length of the longer leg in inches.

Mathematics
1 answer:
Ratling [72]3 years ago
8 0
The equation to represent the area of the triangle would be:
y = 1/2(x²) - (7/2)x

The equation to represent the perimeter of the triangle would be:
y = 3x - 6

The solutions to the system would be (12, 30) or (1, -3).  The only viable solution is (12, 30).

Explanation
The area of a triangle is found using the formula
A = 1/2bh

For our triangle, b = x and h = x-7, so we have:
A = 1/2(x)(x-7)
A = 1/2(x²-7x)
A = 1/2(x²) - (7/2)x

We will replace A with y, so we have:
y = 1/2(x²) - (7/2)x

The perimeter of a triangle is found by adding together all sides, so we have:
P = (x-7) + x + (x+1)

Combining like terms we get:
P = 3x - 6

We will replace P with y, so we have:
y = 3x - 6

Since both equations have y isolated on one side, it will be easy to use substitution to solve the system:

3x - 6 = 1/2(x²) - (7/2)x

It's easier to work with whole numbers, so we will multiply everything by 2:
6x - 12 = x² - 7x

We want all of the variables on one side, so we will subtract 6x:
6x - 12 - 6x = x² - 7x - 6x
-12 = x² - 13x

When solving quadratics, we want the equation equal to 0, so we will add 12:
-12+12 = x² - 13x + 12
0 = x² - 13x + 12

This is easy to factor, as there are factors of 12 that sum to -13; -12(-1) = 12 and -12+-1 = -13:
0 = (x-12)(x-1)

Using the zero product property, we know that either x-12=0 or x-1=0; therefore x=12 or x=1.

Putting these back into our equation for perimeter (the simplest one) we have:
y = 3(12)-6 = 36-6 = 30; (12, 30);
y = 3(1) - 6 = 3 - 6 = -3; (1, -3)

We cannot have a negative perimeter, so the only viable solution is (12, 30).
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1.5 cups

Step-by-step explanation:

3x2=6

4 goes into 6 once and leaves 2 which is half of 4

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3 years ago
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9. If 15 men working 10 days earn $500. How much will 12 men earn working 14,
Leno4ka [110]

Answer:

Firstly need to understand the questions.

If 15 men working 10 days, They earn 500.

If 12 men working 14 days, how much they will earn?

Step-by-step explanation:

560

Solution:

15 men working for 10 days =500

1 man working for 1 day = 500/(10*15) =10/3

Now 12 men working for 14 days = 12*14*10/3 =560.

I am pretty sure that answer is right.

Thank you :)

6 0
2 years ago
I'd really appreciate some help on these questions, I'm really struggling
nikklg [1K]
We can use linear combinations of the equations to eliminate variables.

3x - 4y = 1
-2x + 3y = 1

To eliminate y we'll make the linear combination of 3 times the first equation minus four times the second. 

9x - 12y = 3
-8x + 12y = 4

Adding,
x = 7

We could solve for y directly but let's use another linear combination, twice the first plus three times the second:

2(3x - 4y) + 3(-2x + 3y)= 2(1)+3(1)

y = 5

Check: 3(7)-4(5)=1 good.   -2(7)+3(5)=1 good.

Q18 Answer: (7,5)

y = -3x + 5
5x - 4y = -3

4y +1(5x - 4y) = 4(-3x + 5) + 1(-3)

5x = -12x + 20 -3

17 x = 17

x = 1
y = -3(1) + 5 = 2
Check: 5(1) - 4(2) = -3 good

Q19 Answer (1,2)

6x + 5y = 25
x = 2y + 24
6x = 12y + 144
5y = 25 - 12y - 144
17y = -119
y = -119/17= -7
x = 2y+24= 10
Check: 6(10)+5(-7)=25 good   2y+24=2(-7)+24=10=x good

Q20 Answer (10,-7)


3x + y = 18
-7x + 3y = -10
9x  + 3y = 54
9x - -7x = 54 - -10
16x = 64
x=4
y = 18 -3x = 18-12=6

Check: 3(4)+6=18 good,  -7(4)+3(6)=-10 good

Q21 Answer: (4,6)

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There are 30 books on the library shelf, Sam noted that 1/6 of them are non fiction. Dan notes that 3/5 of the books are fiction
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Answer:

7

Step-by-step explanation:

Fiction books:  (1/6) of 30 books = 5 books

Non-fiction books:  (3/5) of (30) books = 18 books

The subtotal here is 5 + 18, or 23.  

23 of the books are either fiction or non fiction.

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Answer:

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Step-by-step explanation:

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8x/8x = -7/8

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