A toy manufacturer needs a piece of plastic in the shape of a right triangle with the longer leg 1 cm more than the shorter leg and the hypotenuse 2 cm more than the shorter leg. how long should the sides of the triangle​ be?

Solution:

we have been asked to find the average of the first n positive even numbers.

Let the firm n positive even numbers be

[tex] 2,4,6,8,10,12,.........2(n-1), 2n [/tex]

As we know that average of numbers[tex] =\frac{\text{Sum of Numbers }}{Total number of Numbers}\\ [/tex]

Required Average[tex] =\frac{2+4+6+2(n-1)+2n}{n}\\ [/tex]

Required Average[tex] =\frac{2(1+2+3+....+(n-1)+n)}{n} [/tex]

As we know the sum of first n terms[tex] =\frac{n(n+1)}{2} [/tex]

Required Average[tex] =\frac{2n(n+1)}{2n}=n+1 \\ [/tex]

To graph the inequality 2x - 5y ≤ 6, plot the line 2x - 5y = 6 with a solid boundary, test a point to determine the solution region, and shade the region that satisfies the inequality, which is below and to the right of the boundary line.

To graph the inequality 2x - 5y ≤ 6, you must first draw the boundary line of the inequality. This line is created by graphing the equation 2x - 5y = 6.

Start by finding the intercepts. For the x-intercept, set y to 0 and solve for x, which gives x = 3. For the y-intercept, set x to 0 and solve for y, which gives y = -6/5. These two points can be used to draw the boundary line. Since the inequality is '≤', this line will be solid to show that points on the line satisfy the inequality.

Now you must determine which side of the line contains the solutions to the inequality. You can do this by choosing a test point that is not on the line, like (0,0). Substitute this point into the inequality: 2(0) - 5(0) ≤ 6 is true, so the area containing (0,0), which is below and to the right of the line, is the solution region. Shade this area to show all points that satisfy the inequality 2x - 5y ≤ 6.

Answer:

solution is (6,-1)

Step-by-step explanation:

[tex]2x + 6y = 6[/tex]

[tex]3x – 2y = 20[/tex]

WE use elimination method to solve for x  and y

Multiply the second equation by 3

[tex]3x – 2y = 20[/tex] times 3

[tex]9x – 6y = 60[/tex]

[tex]2x + 6y = 6[/tex] , add both equations

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[tex]11x = 66[/tex]

Divide both sides by 11

[tex]x=6[/tex]

Plug in 6 for x in first equation

[tex]2x + 6y = 6[/tex]

[tex]2(6) + 6y = 6[/tex]

[tex]12 + 6y = 6[/tex], subtract 12 on both sides

[tex]6y = -6[/tex]

divide both sides by 6

[tex]y=-1[/tex]

[tex]x=6  \ and \ y= -1[/tex]

A quadratic function seems to be a second-order polynomial function. Another quadratic function has the following general form:

here,

and,

The quadratic formula is obtained by completing the square on some kind of quadratic equation instead in the standard format. Because the largest power equals two, it is indeed a 2nd-degree polynomial equation.

As a result, the aforementioned response is correct.

Learn more about quadratic equation here:

https://brainly.com/question/19942005

Answer:

The correct option is B. [tex]y=-3x+4 \ \text{and}\ \ 3y=-9x+12[/tex]

Step-by-step  explanation:

Consider the two linear equation:

[tex]a_1x+b_1y+c_1=0\ \text{and}\ \ a_2x+b_2y+c_2=0[/tex]

The pair of linear equation has infinitely many solutions when:

[tex]\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}[/tex]

Now consider the option.

Option a) [tex]y=-3x+4 \ \text{and}\ \ y=-3x-4[/tex]

The above equation can be written as:

[tex]y+3x-4=0 \ \text{and}\ \ y+3x+4=0[/tex]

Therefore,

[tex]\frac{1}{1}=\frac{3}{3}\neq\frac{-4}{4}[/tex]

Thus, this pair has no solution.

Now consider the option B. [tex]y=-3x+4 \ \text{and}\ \ 3y=-9x+12[/tex]

The above equation can be written as:

[tex]y+3x-4=0 \ \text{and}\ \ 3y+9x-12=0[/tex]

Therefore,

[tex]\frac{1}{3}=\frac{3}{9}=\frac{-4}{-12}[/tex]

Thus, this pair has infinitely many solution.

Hence, the correct option is B. [tex]y=-3x+4 \ \text{and}\ \ 3y=-9x+12[/tex]

Solution:

we are given that

In a class, every student knows French or German (or both).

15 students know French, and 17 students know German.

Suppose there are x student who knows both French and German.

Then Total number of student in the class will be [tex] (15+17-x)=32-x [/tex]

But to guess the largest possible number of student in the class we can assume x=0

Hence the largest Possible number of Student in the class=32

Answer:

[tex]y=-1[/tex]

Step-by-step explanation:

We have been given an equation of parabola [tex]x^2=4y[/tex]. We are asked to find the directrix of our given parabola.    

First of all, we will divide both sides of our given equation by 4.

[tex]\frac{x^2}{4}=\frac{4y}{4}[/tex]      

[tex]\frac{x^2}{4}=y[/tex]      

[tex]y=\frac{x^2}{4}[/tex]      

Now, we will compare our equation with vertex form of parabola:

[tex]y=a(x-h)^2+k[/tex], where, (h,k) represents vertex of parabola.

We can see that the value of a is [tex]\frac{1}{4}[/tex], [tex]h=0[/tex] and [tex]k=0[/tex].

Now, we will find distance of focus from vertex of parabola using formula [tex]p=\frac{1}{4a}[/tex].

Substituting the value of a in above formula, we will get:

[tex]p=\frac{1}{4*\frac{1}{4}}[/tex]

[tex]p=\frac{1}{1}=1[/tex]

We know that directrix of parabola is [tex]y=k-p[/tex].

Substituting the value of k and p in above formula, we will get:

[tex]y=0-1[/tex]

[tex]y=-1[/tex]

Therefore, the directrix of our given parabola is [tex]y=-1[/tex].

Final answer:

Zia's age is 12 and Raj's age is 15

Explanation:

To solve this problem, we can set up two equations using the given information. Let's assume Zia's age is x. Since Raj is 3 years older than Zia, Raj's age can be represented as x + 3.

The first equation is [tex]x^2 + (x + 3)^2 = 369.[/tex]

Simplifying this equation, we get [tex]x^2 + x^2 + 6x + 9 = 369.[/tex]

Combining like terms, we have [tex]2x^2 + 6x + 9 = 369.[/tex]

Next, we can rearrange the equation by subtracting 369 from both sides and combining like terms to get [tex]2x^2 + 6x - 360 = 0.[/tex]

Now, we can solve this quadratic equation for x using factoring, completing the square, or the quadratic formula. The solutions are x = -15 and x = 12.

Since age cannot be negative, Zia's age is 12 and Raj's age is 12 + 3 = 15.

"The correct answer is B. 79.

To solve the given problem, we need to calculate 15 4/5 percent of 50. First, let's convert the mixed number to an improper fraction to make the calculation easier.

 15 4/5% as a fraction is 15 + 4/5, which can be written as 75/5 + 4/5 = 79/5.

Now, to find the percentage of 50, we multiply 50 by the fraction representing the percentage:

[tex]\[ \frac{79}{5} \times 50 = \frac{79 \times 50}{5} \][/tex]

Next, we simplify the multiplication:

[tex]\[ \frac{79 \times 50}{5} = \frac{3950}{5} \][/tex]

Now, we divide 3950 by 5:

[tex]\[ \frac{3950}{5} = 790 \][/tex]

However, since we are looking for 15 4/5 percent and not 15 4/5 times 50, we need to adjust our calculation by dividing by 100 to get the correct percentage:

[tex]\[ \frac{790}{100} = 79 \][/tex]

Therefore, the final answer is B. 79."

To find the distance the ball traveled after it bounces off the wall, we can use the proportion 34/x = 20/16, which simplifies to x = 27.2 inches.

To find the proportion that can be used to find the distance, x, the ball traveled after it bounces off the wall, we can use the equation:

34/x = 20/16

After cross multiplying, we get:

34 × 16 = 20x

Simplifying further:

544 = 20x

Dividing both sides by 20, we find:

x = 27.2

Therefore, the distance the ball traveled after it bounces off the wall is 27.2 inches.

To factor the quadratic expression x² -3x -40, we find that the expression factors into (x - 8)(x + 5).

The question involves factoring the quadratic expression x² -3x -40 completely. To factor this expression, we look for two numbers that multiply to -40 and add to -3. The numbers that meet these conditions are -8 and 5, since

(-8) * 5 = -40 and

(-8) + 5 = -3.

Therefore, the expression can be factored into (x - 8)(x + 5).

Final answer:

To estimate the total cost, round each item to the nearest $0.10, sum them up, and subtract from the amount given by the customer. The estimated total cost is $9.80, and the estimated change to be returned is $1.45.

Explanation:

To estimate the total amount the customer owes after rounding to the nearest $0.10, we need to perform the following calculations:

Apple pie: $8.04, rounded to $8.00

6 cloves of garlic: $0.09 each, so $0.09 x 6 = $0.54, rounded to $0.50

Quart of milk: $1.31, rounded to $1.30

The estimated total cost is $8.00 (apple pie) + $0.50 (garlic) + $1.30 (milk) = $9.80.

The customer gives a total of $5 + $6 + $0.25 = $11.25.

Subtracting the estimated total cost from the amount given by the customer:
$11.25 - $9.80 = $1.45, which is the estimated change to be returned to the customer.

Answer:4

Step-by-step explanation:just did it on edge

To solve log6(x) + log6(x + 5) = 2, combine the logarithms, rewrite in exponential form, and solve the resulting quadratic equation. The solutions are x = -9 and x = 4, but since logarithms of negative numbers are undefined, the final answer is x = 4.

To solve this logarithmic equation, follow these steps:

Final Answer: x = 4.

Final answer:

To sum the given polynomials, you combine like terms, resulting in the final expression of -3x^2 + 6.

Explanation:

The sum of the polynomials 2x2 + 3x - 4, 8 - 3x, and -5x2 + 2 can be found by combining like terms. This process involves adding the coefficients of terms with the same power of x.

Step 1: Identify and arrange like terms.

Step 2: Combine like terms.

Step 3: Write the sum of the polynomials.
The sum is: -3x2 + 6.