Two sides of an acute triangle measure 5 inches and 8 inches. The length of the longest side is unknown/ What is the greatest possible whole-number length of the unknown side?
8 inches
9 inches
12 inches
13 inches

Answer:  [tex]8\sqrt{2}[/tex]

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Work Shown:

x = length of each leg

Use the pythagorean theorem to get...

[tex]a^2+b^2 = c^2\\\\x^2+x^2 = 16^2\\\\2x^2 = 256\\\\x^2 = 256/2\\\\x^2 = 128\\\\x = \sqrt{128}\\\\x = \sqrt{64*2}\\\\x = \sqrt{64}*\sqrt{2}\\\\x = 8\sqrt{2}\\\\[/tex]

Or as an alternative path, you can divide the hypotenuse by [tex]\sqrt{2}[/tex] and then rationalize the denominator.

Both of these methods apply to 45-45-90 triangles only.

Answer:

[tex]8\sqrt{2}[/tex]

Step-by-step explanation:

We can sum the angles in a triangle to 180°:

m<D + m<E + m<F = 180°

m<D + 90° + 45° = 180°

m<D = 45°

Therefore, the triangle is an isosceles right triangle, since the base angles, <F and <D are the same measure. Because the triangle is isosceles, the legs are congruent, meaning that

DE = EF

We can use the Pythagorean Theorem([tex]a^{2} + b^{2} = c^{2}[/tex]) and plug in 16 as c, and DE and EF as a and b:

[tex]DE^{2} + EF^{2} = 16^{2}[/tex]

Since DE is the same as EF, we can substitute it in:

[tex]DE^{2} + DE^{2} = 256\\2DE^{2} = 256\\DE^{2} = 128\\DE = \sqrt{128}[/tex]

The square root of 128 is the same as [tex]8\sqrt{2}[/tex] since 128 can be factored into 64 x 2. So [tex]8\sqrt{2}[/tex] is the answer.

If you are looking for an easier way to do this, once you are able to recognize DEF is an isosceles triangle, you can say [tex]DE \times \sqrt{2} = 16[/tex] as this applies to all isosceles right triangles. This simplifies to:

[tex]DE = \frac{16}{\sqrt{2}} \\DE = \frac{16}{\sqrt{2} } \times \frac{\sqrt{2}}{\sqrt{2} } \\DE = \frac{16\sqrt{2} }{2} \\DE = 8\sqrt{2}[/tex]

There are several ways

Let's check all

The limit of the given expression lim x → 0 [4eˣ/2 − 4 − 2x − 12x² +  x³] is determined as -2.

The limit of the given expression is calculated as follows;

lim x → 0 [4eˣ/2 − 4 − 2x − 12x² +  x³]

= (4e⁰)/2 - 4 - 2(0) - 12(0) + (0)

= (4)/2 - 4

= 2 - 4

= - 2

Thus, the limit of the given expression lim x → 0 [4eˣ/2 − 4 − 2x − 12x² +  x³] is determined as -2.

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Answer: The correct answer is A, [tex](-2)^{2}[/tex].

Step-by-step explanation:

A negative multiplied with another negative is a positive. Squaring is multiplying an integer by itself. Thus, it would look like -2×-2, which is equal to 4. I hope this helps!! :)

Answer: (2,-2)

Step-by-step explanation:

Rotating 90 degrees counterclockwise (about the origin) maps (x,y) onto (-y,x), so the answer is N'(2,-2)

Multiplication is the process of multiplying, therefore, adding a number to itself for the number of times stated. The correct option is B.

Multiplication is the process of multiplying, therefore, adding a number to itself for the number of times stated. For example, 3 × 4 means 3 is added to itself 4 times, and vice versa for the other number.

When the indicated operation 4R₁ → R₁ is applied then the matrix will be,

[tex]\left[\begin{array}{ccc}1&2&0\\-2&-1&3\end{array}\right] \rightarrow \left[\begin{array}{ccc}4&8&0\\-2&-1&3\end{array}\right][/tex]

Hence, the correct option is B.

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The algebraic expression is (7 + 3m)/27 < n

The statement is given as:

"The quotient of seven more than three times a number m and the number 27 is less than n"​

The numbers are represented as n and m

Quotient means divide

So, we have:

The quotient of seven more than three times a number m and the number 27 is represented as:

(7 + 3m)/27

The result is less than n.

So, we have:

(7 + 3m)/27 < n

Hence, the algebraic expression is (7 + 3m)/27 < n

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Let's replace every copy of x with 7 and simplify.

[tex]\sqrt{5x+1}+9 = 3\\\\\sqrt{5*7+1}+9 = 3\\\\\sqrt{35+1}+9 = 3\\\\\sqrt{36}+9 = 3\\\\6+9 = 3 \ \text{ .... note the 6 on the left isn't negative}\\\\15 = 3\\\\[/tex]

Which is clearly false. This is probably the set of steps you followed to get "no solution".

Why is there no solution? Well let's subtract 9 from both sides to isolate the square root

[tex]\sqrt{5x+1}+9 = 3\\\\\sqrt{5x+1}+9-9 = 3-9\\\\\sqrt{5x+1} = -6\\\\[/tex]

Recall that the result of a square root operation is never negative. The range of [tex]y = \sqrt{x}[/tex] and [tex]y = \sqrt{5x+1}[/tex] is the set of nonnegative numbers. There is no way we can have the left hand side result in -6

If there was a negative sign out front the square root and we have this instead

[tex]-\sqrt{5x+1}+9 = 3\\\\[/tex]

then the answer would be x = 7

Unfortunately, there isn't such a negative sign, so we stick with "no solutions".

Answer:   [1, infinity)

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Explanation:

Let's isolate x in the first inequality mentioned.

x + 7 ≥ 4

x + 7-7 ≥ 4-7

x ≥ -3

I subtracted 7 from both sides to undo the +7.

Now isolate x in the second inequality. We first add 8 to both sides, then divide both sides by 11.

11x - 8 ≥ 3

11x - 8+8 ≥ 3+8

11x ≥ 11

11x/11 ≥ 11/11

x ≥ 1

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The two solutions we found were

x ≥ -3 and x ≥ 1

Next, draw out a number line with closed circles at -3 and 1. Shade to the right of each closed circle endpoint as you see below.

Notice that the two inequalities overlap for x ≥ 1

In other words, if a number is -3 or larger AND 1 or larger, then the number is 1 or larger.

The two solutions intersect to form x ≥ 1

The inequality x ≥ 1 is the same as 1 ≤ x which is the same as 1 ≤ x < infinity

Then that turns into the interval notation of [1, infinity)

The square bracket includes the endpoint 1.

If needed, use the infinity symbol in place of the word "infinity".

Answer:

[1, ∞)

Step-by-step explanation:

To solve a compound inequality, first separate it into two inequalities and solve them separately:

Solution of Inequality 1:

x + 7 ≥ 4

⇒ x ≥ 4 - 7

⇒ x ≥ -3

Solution of Inequality 2:

11x - 8 ≥ 3

⇒ 11x ≥ 3 + 8

⇒ 11x ≥ 11

⇒ x ≥ 1

As both graphs overlap for x ≥ 1, this is the solution.

Therefore, in interval notation:  [1, ∞)

If he is not eligible for Roth IRAs, then he cannot contribute to Roth IRAs. However, he can contribute $3,330 monthly under his company's Matching Program.

The Individual Retirement Account to which employees in the United States can contribute after-tax earnings is called the Roth IRA.

Under its governing principles, certain individuals whose annual income exceeds stipulated amounts are INELIGIBLE to contribute.

Recall that the question indicates that his company offers a matching program with a ratio 1:1 up to 3% of his annual salary.

Since his annual salary is $111, 000,

3% x 111,000 = 3,330.

This means that annually, the total matching contribution due to Adrian would be:

3,330 x 2 = 6, 660.

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Answer:

Step-by-step explanation:

Comment

Answer (-4,5,6)

B

Answer:

Step-by-step explanation:

Solution

The first step is to simplify the top of the fraction. Subtract 5.6 - 2.7

The numerator (top) and denominator(bottom) are decimal fractions. They are 2.9 for the top and 2.9 for the bottom.

The whole expression reduces down to 1, which is an integer.

Note

You have choices. If you want exact answers you will have to list those choices.

Answer: 1

The total cost of 35 T-shirts, 45 pairs of shorts, and 25 windbreakers is $973.75.

The unitary method is a method for solving a problem by the first value of a single unit and then finding the value by multiplying the single value.

Cost of 1 T-shirt = $7.50

Cost of 35 T-shirts = 7.50 x 35

                                = 262.5

Cost of 1 Shorts = $9.00

Cost of 45 T-shirt = 9.00 x 45

                             = 405

Cost of 1 windbreakers = $12.25

Cost of 25 windbreakers = 12.25 x 306.25

So, The total cost of 35 T-shirts, 45 pairs of shorts, and 25 windbreakers

= 262.5+405+306.25

=  $973.75

Hence Her total cost is $973.75.

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Answer:

They would have $3 dollars left over.

Step-by-step explanation:

If you divide 187 by 4, you get 46.75. But if you look again, you can see that the remainder is 3 and it can confirm that it is the left over money that the club has now after buying as many ice cream cartons as possible.

Answer:

-12

Step-by-step explanation:

What you have to do is plug -4  into the function as a puzzle. So, replace all x with -4.

f(-4) = 7(-4) - 4(-4) = -28 + 16 = -12

f (-4) = 7x ( 1-4 ) - 4 (-4)

       = -28 + 16

      = -12

An equation is a mathematical statement this is made from two expressions related by an equal sign. for instance, 3x – 5 = 16 is an equation. solving this equation, we get the fee of the variable x as x = 7.

A declaration of the equality of mathematical expressions. An expression representing a chemical reaction by using chemical symbols. equation.

Conclusion: There are three major forms of linear equations: point-slope form, standard form, and slope-intercept form. We review all three in this article.

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Answer:

The prime factorization of 14 is 2 x 7.

Step-by-step explanation:

The prime factorization of 14 is 2 x 7.

Answer:

14 is a composite number

Step-by-step explanation:

A function assigns the values. The function will move down by 3 units.

A function assigns the value of each element of one set to the other specific element of another set.

The graph of the function is given below. Now if 3 is replaced by a 0, then the y-intercept of the function will become zero while keeping the slope the same. Therefore, the function will move down by 3 units.

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Using the zero product property, we can set each factor equal to zero.

First Factor [tex]x^{4}+5x^{2}-36[/tex]

[tex]x^{4}+5x^{2}-36=0\\(x^{2}+9)(x^{2}-4)=0\\x^{2}+9=0, x^{2}-4=0\\x=\pm 3i, x=\pm 2[/tex]

Second Factor [tex]2x^{2}+9x+5[/tex]

Using the quadratic formula, we get [tex]x=\frac{-9 \pm \sqrt{41}}{4}[/tex].

Following the 30% rule of thumb, Sally should only spend a maximum of $900 on her rent

Given Data

Let us find 30% of $3000

= 30/100* $3000

= 0.3*$3000

= $900

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[tex]15 \div 3( - 2) + | - 10| \\ 5( - 2) + 10 \\ - 10 + 10 \\ 0[/tex]

Hope it helps

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No, Number 82296 is not a perfect square.

And, We multiply by 82296 in the number 82296 to amke a perfect square.

A square root of a number is a value that multiplied by itself gives the same number.

Given that;

The number is,

⇒ 82296

Now,

If the number 82296 is a perfect square number.

So, We get;

⇒ x² = 82296

⇒ x = √82296

⇒ x = 286.87

So, This number is not a perfect square number.

If we multiply by the number 82296 in the number 82296, we get;

⇒ 82296 × 82296

⇒ (82296)²

Thus, It become a perfect square number.

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Answer:

y = 15/-4 + 2.5x

Step-by-step explanation:

10x - 4y = 15

-4y = 15 - 10x

y = 15/-4 + 2.5x

Answer:  [tex]7\text{Ln}\left(e^{2x}+10\right)+C[/tex]

This is the same as writing 7*Ln( e^(2x) + 10) + C

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Explanation:

Start with the equation [tex]u = e^{2x}+10[/tex]

Apply the derivative and multiply both sides by 7 like so

[tex]u = e^{2x}+10\\\\\frac{du}{dx} = 2e^{2x}\\\\7\frac{du}{dx} = 7*2e^{2x}\\\\7\frac{du}{dx} = 14e^{2x}\\\\7du = 14e^{2x}dx\\\\[/tex]

The "multiply both sides by 7" operation was done to turn the 2e^(2x) into 14e^(2x)

This way we can do the following substitutions:

[tex]\displaystyle \int \frac{14e^{2x}}{e^{2x}+10}dx\\\\\\\displaystyle \int \frac{1}{e^{2x}+10}14e^{2x}dx\\\\\\\displaystyle \int \frac{1}{u}7du\\\\\\\displaystyle 7\int \frac{1}{u}du\\\\\\[/tex]

Integrating leads to

[tex]\displaystyle 7\int \frac{1}{u}du\\\\\\7\text{Ln}\left(u\right)+C\\\\\\7\text{Ln}\left(e^{2x}+10\right)+C\\\\\\[/tex]

Be sure to replace 'u' with e^(2x)+10 since it's likely your teacher wants a function in terms of x. Also, do not forget to have the plus C at the end. This is a common mistake many students forget to do.

To verify the answer, you can apply the derivative to it and you should get back to the original integrand of [tex]\frac{14e^{2x}}{e^{2x}+10}[/tex]

The amount that the home will be worth after 7 years is  $457763.67

The geometric sequence is sometimes called the exponential sequence. The formula for calculating the nth term of an arithmetic sequence is expressed as:

Tn = ar^n-1

where

a is the first term = 120,000

r is the common ratio = 150000/120000 = 1.25

Required

7th term

Substitute

T7 = 120,000(1.25)^7-1
T7 = 120,000(1.25)^6

T7 = 120,000(3.8147)

T7 = $457763.67

Hence the amount that the home will be worth after 7 years is  $457763.67

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Answer:

(c) $457,763.67

Step-by-step explanation:

hope this helps:)

For the given function h(x), we have:

a) at x = -2 and x = 2.

b) y = 0 and y = 3.

First, we want to get the values of x at which we have maximums. To do that, we need to see the value in the horizontal axis at where we have maximums.

By looking at the horizontal axis, we can see that the maximums are at:

x = -2 and at x = 2.

Now we want to get the maximum values, to do that, we need to look at the values in the vertical axis.

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Answer:

A because it shows that the calculations are correct as well as combination pattem