A sector of a circle is shown. Find its area, rounded to the nearest tenth.
a)9.8 cm^2
b)10.5 cm^2
c)11.2 cm^2
d)12.3 cm^2 ...?

To prove the given equation involving tangent, we use trigonometric identities and simplification techniques to show that both sides of the equation are equal.

To prove the given equation: tan(π/4 x) - tan(π/4-x)/tan(π/4 x) tan(π/4-x) = 2sin(x)cos(x).

We can simplify this equation using the trigonometric identities for tan:

By substituting these identities, we get:

1 - 1/(tan(π/4-x)tan(π/4 x)) = 2sin(x)cos(x)

From here, we can simplify further and use the following trigonometric identities:

Using these identities, we can simplify the equation to:

-2sin(x)cos(x) = -2sin(x)cos(x)

Therefore, the given equation is proved to be true.

Answer:

the answer is C.

Step-by-step explanation:

Answer: C 39 cm

Step-by-step explanation:

Answer:

7^4

Step-by-step explanation:

Final answer:

To round 16.2 to the nearest whole number, check the first digit after the decimal. It's less than 5, so round down to 16.

Explanation:

When rounding 16.2 to the nearest whole number, you need to look at the first digit after the decimal point, which is 2 in this case. If that digit is 5 or greater, you would round up. If it is less than 5, you round down. Since 2 is less than 5, we round down, so the whole number that 16.2 gets rounded to is 16.

The answer is provided in the image attached.

The horizontal component of the velocity of a ball thrown at a 30 degree angle above the horizontal with an initial speed of 10 ft/s will remain approximately 8.66 ft/s after 0.50s, assuming no air resistance.

A ball is thrown at a 30 degree angle above the horizontal with a speed of 10 ft/s. After 0.50s the horizontal component of its velocity will remain the same, assuming no air resistance, because horizontal velocity in projectile motion is constant if air resistance is ignored. The vertical component will change due to gravity, but that does not affect the horizontal component.

To find the horizontal component of the velocity, we can use the cosine of the launch angle:
Horizontal velocity (v_x) = initial speed * cos(launch angle)
So, v_x = 10 ft/s * cos(30 degrees)

= 10 ft/s * (√3/2)

≈ 8.66 ft/s

As there is no horizontal acceleration in ideal projectile motion, after 0.50s, the horizontal component of the ball's velocity will still be approximately 8.66 ft/s.

The lengths of the sides of the triangle ABC is AB<AC<BC. Therefore, option 4 is the correct answer.

If one side of a triangle is longer than another side, then the angle opposite the longer side will be larger than the angle opposite the shorter side.

Given that, in triangle ABC, m∠A = 110°, m∠B = 45° and m∠C = 25°.

Opposite to m∠A is side BC, opposite to m∠B is side AC and opposite to m∠C is side AB.

Here, largest angle is ∠A, so the longer side is BC and

Smaller angle is ∠C, so the shortest side is AB and

Now, AB<AC<BC

Therefore, option 4 is the correct answer.

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Final answer:

To find the uniform width of the deck around a 4 yd by 5 yd rectangular pool, with the combined area of the deck and pool being 90 square yards, we form a quadratic equation based on the dimensions. After solving the equation, we obtain the width of the deck as the positive solution.

Explanation:

The student has a rectangular pool and wants to build a concrete deck of a uniform width around it such that the combined area of the pool and deck is 90 square yards. To solve this problem, we first need to recognize that the area of just the pool is length times width, which is 4 yd by 5 yd. This gives us an area of 20 square yards for the pool alone.

Next, we consider that the deck surrounds the pool uniformly. If we let 'w' be the width of the deck, then the new length and width of the pool plus the deck becomes (4 + 2w) yd and (5 + 2w) yd respectively, since the deck is added to all four sides of the pool.

The combined area is therefore (4 + 2w) * (5 + 2w) = 90 square yards. Expanding the left side of the equation and simplifying, we get 20 + 18w + 4w2 = 90. This simplifies further to 4w2 + 18w - 70 = 0 when we subtract 90 from both sides.

To solve for 'w', we factor the quadratic equation or use the quadratic formula. Upon factoring, we find that w could have two potential values (one positive and one negative), but since width cannot be negative, we discard the negative value. The positive value gives us the width of the deck.

-5.77 is the answer

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Final answer:

The expression 2.5 – 3.1 – 3.86 + 2.7 – 4.01 simplifies to -5.77 by adding the positive numbers together, adding negative numbers together, and then combining the two results.

Explanation:

To simplify the expression 2.5 – 3.1 – 3.86 + 2.7 – 4.01, we need to perform the subtraction and addition operations in sequence.

The simplified result of the expression is -5.77.

Answer:

No; corresponding angles are not congruent

Step-by-step explanation:

we know that

[tex]124\°[/tex] and [tex]118\°[/tex] are corresponding angles

The lines l and m are not parallel

because

[tex]124\°\neq 118\°[/tex]

Answer: 1) Value is placed on top of the X =90

2) Value is placed on the bottom of the X =21

3)  Factored form will be (3x+5)(3x+2)

Step-by-step explanation:

Since we have given that

[tex]9x^2 + 21x + 10[/tex]

As we know the quadratic form :

[tex]ax^2+bx+c=0[/tex]

1) Value is placed on top of the X is given by

[tex]a\times c=9\times 10=90[/tex]

2) Value is placed on the bottom of the X is given by

[tex]b=21[/tex]

3) Factorised form:

We will use "Split the middle term":

[tex]9x^2 + 21x + 10\\\\=9x^2+15x+6x+10\\\\=3x(3x+5)+2(3x+5)\\\\=(3x+2)(3x+5)[/tex]

Hence, Factored form will be

(3x+5)(3x+2)

The rate of change in miles per gallon for a car trip represents the number of miles the car can travel per gallon of gasoline used. The ordered pairs (0, 2), (1, 2), (2, 3), (1, 4), (3, 1) accurately correspond with the values provided in the given table.

In the given situation, the rate of change of a graph showing the miles per gallon of gasoline a car uses on a trip would represent the number of miles traveled for every one gallon of gas used. This is because the rate of change in this context quantifies how much distance can be covered (in miles) using one unit of gasoline (in gallon). For instance, if the rate of change is 30, it indicates that the car can travel 30 miles using just one gallon of gasoline.

For the second part of your question, matching the ordered pairs with the table, the correct match would be (0, 2), (1, 2), (2, 3), (1, 4), (3, 1). This set of pairs corresponds accurately to the values given in your table for x and y. To understand this, you can match each x value with its corresponding y value in the table. For example, when x is 0, y is 2 - and this gives you the ordered pair (0, 2).

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The average rate of change of f(x) over the interval [-1, 1] is 5.

To find the average rate of change of f(x) over the interval [-1, 1], we use the formula:

[tex]\[ \text{Average Rate of Change} = \frac{\text{Change in } f(x)}{\text{Change in } x} \][/tex]

First, we find the change in f(x) over the interval [-1, 1]. The function values at x = -1 and x = 1 are f(-1) = -5 and f(1) = 5 respectively. So, the change in f(x) is 5 - (-5) = 10.

Next, we find the change in x over the same interval. The interval [-1, 1] has a width of 1 - (-1) = 2.

Finally, we divide the change in f(x) by the change in x to get the average rate of change:

[tex]\[ \text{Average Rate of Change} = \frac{10}{2} = 5 \][/tex]

So, the average rate of change of f(x) over the interval [-1, 1] is 5.

Answer: 1) C. (x+3)(x+7)

2) D. (x-3)(x-8)

3) C. (x-2)(x+4)

5) B. (x+9)(x-9)

Step-by-step explanation:

1)

x2 + 10x + 21

(we look for 2 numbers whose product gives 21 and whose sum gives 10, then replace 10x by those two numbers)

x2 + 7x + 3x +21

x(x+7) + 3(x+7)

(x+3)(x+7)

2)

x2 - 11x + 24

(we look for 2 numbers whose product gives 24 and whose sum gives -11, then replace -11x by those two numbers)

x2 -3x-8x +24

x(x-3) -8(x-3)

(x-3)(x-8)

3)

x2 + 2x - 8

(we look for 2 numbers whose product gives -8 and whose sum gives 2, then replace 2x by those two numbers)

x2 + 2x - 8

x2 + 4x - 2x - 8

x(x+4) -2(x+4)

(x+4)(x-2)

5)

x2 - 81

Its a difference of two squares

(x-9)(x+9)