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3 years ago

Sofie makes and sells scarves. Her profit depends on what price she charges for a scarf

Mathematics

1 answer:

Mamont248 [21]3 years ago

5 0

Answer:

a) 5<x<25

b) x≤5 or x≥25

c) x≥5 and x≤25

Step-by-step explanation:

Given the expression the

(x - 5) (50 - 2x) to represent sofie profit based on the price per scarf, x.

a) If Sofie's profit is increasing, them the expression for her profit must be greater than zero as shown.

(x - 5) (50 - 2x)> 0

Solving the inequality:

x-5>0 and 50-2x>0

If x-5>0

x > 5 or 5<x... (1)

Similarly when 50-2x>0

50-2x>0

50>2x

Dividing both sides by 2:

50/2>2x/2

25>x or x<25 ...(2)

Combining 1 and 2

5<x<25

b) if Sofie's profit is decreasing, the profit function must be less than or equal to zero as shown;

(x - 5) (50 - 2x)≤0

Solving the inequality

x-5≤0 and 50-2x≤0

x≤5 and 50≤2x

x≤5 or 25≤x

x≤5 or x≥25

c) If Sofia is not making profit then, the function (x - 5) (50 - 2x) is equal to zero i.e

(x - 5) (50 - 2x) = 0

x - 5 = 0 or 50-2x =0

x = 5 and 50 = 2x

x = 5 and x = 25

x≥5 and x≤25

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If the vertex of a parabola is (-4,-7), what is the axis of symmetry?

mel-nik [20]

Answer:

-4

Step-by-step explanation:

Given vertex, the axis of symmetry is x = h.

4 0

2 years ago

For all integer values of x and constant k, if (x-k)(x-5)=x^2-9x+20, what is the value of K?

amid [387]

<h3>Answer: C) 4</h3>

============================

Work Shown:

Expand out the left hand side

(x-k)(x-5)

x(x-5)-k(x-5)

x^2-5x-kx+5k

Note that the constant term here is 5k. Yes k seems like a variable, but it's actually a constant. Once we know k, we can replace it to get a fixed number. In this case, k = 4 since 5k = 5*4 = 20 to have it match with the 20 at the end of x^2-9x+20

For the x terms, we have -5x-kx = -5x-4x = -9x which matches with the middle term of x^2-9x+20

Therefore,

(x-k)(x-5) = x^2-9x+20

updates to

(x-4)(x-5) = x^2-9x+20

The -4 and -5 multiply to 20, and they also add to -9. This helps confirm we have the right k value.

4 0

4 years ago

Select the three number bonds that represent factor pairs of the number 56. CLEAR CHECK Number bond: 1 times 56 = 56. Number bon

Ray Of Light [21]

Answer:

The answer is -

Number bond: 1 times 56 = 56.

Number bond: 4 times 14 = 56.

Number bond: 7 times 8 = 56.

Step-by-step explanation:

We can define a factor pair as a set of two factors, which, when multiplied together, give a particular product.

For example -

5 and 6 are factors of 30 because 5×6 = 30

Now,

Given -

Number bond: 1 times 56 = 56. ...................(1)

Number bond: 4 times 14 = 56. ...................(2)

Number bond: 5 times 6 = 56. ...................(3)

Number bond: 7 times 8 = 56. ...................(4)

In equation (1)

1 times 56 = 1×56 = 56

⇒1 and 56 are the factor pairs of 56

Satisfies

In equation (2)

4 times 14 = 4×14 = 56

⇒4 and 14 are factor pairs of 56

Satisfies

In equation (3)

5 times 6 = 5×6 = 30 ≠ 56

⇒5 and 6 are not factor pairs of 56

Not Satisfies

In equation (4)

7 times 8 = 7×8 = 56

⇒7 and 8 are factor pairs of 56

Satisfies

∴ we get

The answer is -

Number bond: 1 times 56 = 56.

Number bond: 4 times 14 = 56.

Number bond: 7 times 8 = 56.

6 0

3 years ago

Use (a) the midpoint rule and (b) simpson's rule to approximate the below integral. ∫ x^2sin(x) dx with n = 8.

MaRussiya [10]

Answer:

midpoint rule = 5.93295663

simpson's rule = 5.869246855

Step-by-step explanation:

a) midpoint rule

$\int\limits^b_a {(x)} \, dx$ ≈ Δ x (f(x₀+x₁)/2 + f(x₁+x₂)/2 + f(x₂+x₃)/2 +...+ f(x $_{n}$ _₂+x $_{n}$ _₁)/2 +f(x $_{n}$ _₁+x $_{n}$ )/2)

Δx = (b − a) / n

We have that a = 0, b = π, n = 8

Therefore

Δx = (π − 0) / 8 = π/8

Divide the interval [0,π] into n=8 sub-intervals of length Δx = π/8 with the following endpoints:

a=0, π/8, π/4, 3π/8, π/2, 5π/8, 3π/4, 7π/8, π = b

Now, we just evaluate the function at these endpoints:

$f(\frac{x_{0}+x_{1} }{2} ) = f(\frac{0+\frac{\pi}{8} }{2} ) = f(\frac{\pi }{16})=\frac{\pi^{2}sin(\frac{\pi }{16}) }{256} =$ 0.00752134

$f(\frac{x_{1}+x_{2} }{2} ) = f(\frac{\frac{\pi }{8} +\frac{\pi}{4} }{2} ) = f(\frac{3\pi }{16})=\frac{9\pi ^{2} sin(\frac{3\pi }{16}) }{256}$ = 0.19277080

$f(\frac{x_{2}+x_{3} }{2} ) = f(\frac{\frac{\pi }{4} +\frac{3\pi}{8} }{2} ) = f(\frac{5\pi }{16})=\frac{25\pi ^{2} sin(\frac{5\pi }{16}) }{256}$ = 0.80139415

$f(\frac{x_{3}+x_{4} }{2} ) = f(\frac{\frac{3\pi }{8} +\frac{\pi}{2} }{2} ) = f(\frac{7\pi }{16})=\frac{49\pi ^{2} sin(\frac{7\pi }{16}) }{256}$ = 1.85280536

$f(\frac{x_{4}+x_{5} }{2} ) = f(\frac{\frac{\pi }{2} +\frac{5\pi}{8} }{2} ) = f(\frac{9\pi }{16})=\frac{81\pi ^{2} sin(\frac{7\pi }{16}) }{256}$ = 3.062800704

$f(\frac{x_{5}+x_{6} }{2} ) = f(\frac{\frac{5\pi }{8} +\frac{3\pi}{4} }{2} ) = f(\frac{11\pi }{16})=\frac{121\pi ^{2} sin(\frac{5\pi }{16}) }{256}$ = 3.878747709

$f(\frac{x_{6}+x_{7} }{2} ) = f(\frac{\frac{3\pi }{4} +\frac{7\pi}{8} }{2} ) = f(\frac{13\pi }{16})=\frac{169\pi ^{2} sin(\frac{3\pi }{16}) }{256}$ = 3.61980731

$f(\frac{x_{7}+x_{8} }{2} ) = f(\frac{\frac{7\pi }{8} +\pi }{2} ) = f(\frac{15\pi }{16})=\frac{225\pi ^{2} sin(\frac{\pi }{16}) }{256}$ = 1.69230261

Finally, just sum up the above values and multiply by Δx = π/8:

π/8 (0.00752134 +0.19277080+ 0.80139415 + 1.85280536 + 3.062800704 + 3.878747709 + 3.61980731 + 1.69230261) = 5.93295663

b) simpson's rule

$\int\limits^b_a {(x)} \, dx$ ≈ (Δx)/3 (f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + 2f(x₄) + ... + 2f( $x_{n-2}$ ) + 4f( $x_{n-1}$ ) + f( $x_{n}$ ))

where Δx = (b−a) / n

We have that a = 0, b = π, n = 8

Therefore

Δx = (π−0) / 8 = π/8

Divide the interval [0,π] into n = 8 sub-intervals of length Δx = π/8, with the following endpoints:

a = 0, π/8, π/4, 3π/8, π/2, 5π/8, 3π/4, 7π/8 ,π = b

Now, we just evaluate the function at these endpoints:

f(x₀) = f(a) = f(0) = 0 = 0

$4f(x_{1} ) = 4f(\frac{\pi }{8} )=\frac{\pi^{2}\sqrt{\frac{1}{2}-\frac{\sqrt{2} }{4} } }{16}$ = 0.23605838

$2f(x_{2} ) = 2f(\frac{\pi }{4} )=\frac{\sqrt{2\pi^{2} } }{16}$ = 0.87235802

$4f(x_{3} ) = 4f(\frac{3\pi }{8} )=\frac{9\pi^{2}\sqrt{\frac{\sqrt{2} }{4}-\frac{{1} }{2} } }{16}$ = 5.12905809

$2f(x_{4} ) = 2f(\frac{\pi }{2} )=\frac{\pi ^{2} }{2}$ = 4.93480220

$4f(x_{5} ) = 4f(\frac{5\pi }{8} )=\frac{25\pi^{2}\sqrt{\frac{\sqrt{2} }{4}-\frac{{1} }{2} } }{16}$ = 14.24738359

$2f(x_{6} ) = 2f(\frac{3\pi }{4} )=\frac{9\sqrt{2\pi^{2} } }{16}$ = 7.85122222

$4f(x_{7} ) = 4f(\frac{7\pi }{8} )=\frac{49\pi^{2}\sqrt{\frac{1}{2}-\frac{\sqrt{2} }{4} } }{16}$ = 11.56686065

f(x₈) = f(b) = f(π) = 0 = 0

Finally, just sum up the above values and multiply by Δx/3 = π/24:

π/24 (0 + 0.23605838 + 0.87235802 + 5.12905809 + 4.93480220 + 14.24738359 + 7.85122222 + 11.56686065 = 5.869246855

7 0

3 years ago

Find the value of X in the isosceles triangle shown below.

BigorU [14]

Answer:

D) Square root of 48

Step-by-step explanation:

We have to split the triangle in half and then multiply that by two for the answer. Then we use a²+b²=c² because the two triangles are right triangles, and we know b=4 and c=8, so a²+(4²)=8², or a²+16=64

a² would have to equal 48. a=√48.

7 0

4 years ago