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3 years ago

Consider the graph of the linear equation 2x-5y=10. What is the x-intercept? What is the y-intercept? What is the slope?

1 answer:

4 0

Hey there!

This linear equation is written in standard form, a form in which it's quite easy to find intercepts for one main reason. This reason revolves around zeros. When you see an x intercept, there's always a zero for the y, and when there's a y you see a zero for the x. Therefore, we can do this to get the intercepts.

For x:

2x-5(0) =10
2x = 10
x = (5,0)

For y:

2(0) -5y = 10
-5y=10
y= (0,-2)

Now, to find the slope, we can easily convert it to slope intercept form:

y = mx+b

Where m is the slope and b is he y intercept.

2x-5y=10
-5y=-2x+10
y = 2/5x-2
(notice how we divided both sides by -5)

Therefore, our slope is 2/5. Hope this helps!

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This is finding exact values of sin theta/2 and tan theta/2. I’m really confused and now don’t have a clue on how to do this, pl

Lostsunrise [7]

First,

tan(θ) = sin(θ) / cos(θ)

and given that 90° < θ < 180°, meaning θ lies in the second quadrant, we know that cos(θ) < 0. (We also then know the sign of sin(θ), but that won't be important.)

Dividing each part of the inequality by 2 tells us that 45° < θ/2 < 90°, so the half-angle falls in the first quadrant, which means both cos(θ/2) > 0 and sin(θ/2) > 0.

Now recall the half-angle identities,

cos²(θ/2) = (1 + cos(θ)) / 2

sin²(θ/2) = (1 - cos(θ)) / 2

and taking the positive square roots, we have

cos(θ/2) = √[(1 + cos(θ)) / 2]

sin(θ/2) = √[(1 - cos(θ)) / 2]

Then

tan(θ/2) = sin(θ/2) / cos(θ/2) = √[(1 - cos(θ)) / (1 + cos(θ))]

Notice how we don't need sin(θ) ?

Now, recall the Pythagorean identity:

cos²(θ) + sin²(θ) = 1

Dividing both sides by cos²(θ) gives

1 + tan²(θ) = 1/cos²(θ)

We know cos(θ) is negative, so solve for cos²(θ) and take the negative square root.

cos²(θ) = 1/(1 + tan²(θ))

cos(θ) = - 1/√[1 + tan²(θ)]

Plug in tan(θ) = - 12/5 and solve for cos(θ) :

cos(θ) = - 1/√[1 + (-12/5)²] = - 5/13

Finally, solve for sin(θ/2) and tan(θ/2) :

sin(θ/2) = √[(1 - (- 5/13)) / 2] = 3/√(13)

tan(θ/2) = √[(1 - (- 5/13)) / (1 + (- 5/13))] = 3/2

3 0

3 years ago

Arectangularplotoflandistobefencedinusingtwokinds of fencing. Two opposite sides will use heavy-duty fencing selling for $3 a fo

svlad2 [7]

Answer:

Dimensions of the rectangular plot will be 500 ft by 750 ft.

Step-by-step explanation:

Let the length of the rectangular plot = x ft.

and the width of the plot = y ft.

Cost to fence the length at the cost $3.00 per feet = 3x

Cost to fence the width of the cost $2.00 per feet = 2y

Total cost to fence all sides of rectangular plot = 2(3x + 2y)

2(3x + 2y) = 6,000

3x + 2y = 3,000 ----------(1)

3x + 2y = 3000

2y = 3000 - 3x

y = $\frac{1}{2}[3000-3x]$

y = 1500 - $\frac{3x}{2}$

Now area of the rectangle A = xy square feet

A = x[ $1500-\frac{3x}{2}$ ]

For maximum area $\frac{dA}{dx}=0$

A' = $\frac{d}{dx}(1500x-\frac{3}{2}x^{2})$ = 0

1500 - 3x = 0

3x = 1500

x = 500 ft

From equation (1),

y = 1500 - $\frac{3}{2}\times 500$

y = 1500 - 750

y = 750 ft

Therefore, for the maximum area of the rectangular plot will be 500 ft × 750 ft.

two fencing 3(500+500) = $3000

other two fencing 2(750+750) = $3000

8 0

3 years ago

Can somebodt please help me

irakobra [83]

can someone help me we out

4 0

4 years ago

Find f'(x) and state the domain of f': f(x) = In (2x^2+1)

-Dominant- [34]

Answer:

$f'(x) = \frac{4x}{2x^2+1}$

Domain: All Real Numbers

General Formulas and Concepts:

Algebra I

Domain is the set of x-values that can be inputted into function f(x)

Calculus

The derivative of a constant is equal to 0

Basic Power Rule:

f(x) = cxⁿ
f’(x) = c·nxⁿ⁻¹

Chain Rule: $\frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)$

Derivative: $\frac{d}{dx} [ln(u)] = \frac{u'}{u}$

Step-by-step explanation:

Step 1: Define

f(x) = ln(2x² + 1)

Step 2: Differentiate

Derivative ln(u) [Chain Rule/Basic Power]: $f'(x) = \frac{1}{2x^2+1} \cdot 2 \cdot 2x^{2-1}$
Simplify: $f'(x) = \frac{1}{2x^2+1} \cdot 4x$
Multiply: $f'(x) = \frac{4x}{2x^2+1}$

Step 3: Domain

We know that we would have issues in the denominator when we have a rational expression. However, we can see that the denominator would never equal 0.

Therefore, our domain would be all real numbers.

We can also graph the differential function to analyze the domain.

5 0

3 years ago

As an estimation we are told 5 miles is 8 km. Convert 32.5 miles to km.

elixir [45]

Answer:

52 km

Step-by-step explanation:

Given, 5 miles is 8km.

So, 1 mile will be (8/5) km.

Now, 32.5 miles will be (8/5) x 32.5 = 52 km

5 0

3 years ago