A ball is thrown downward from the top of a 240​-foot building with an initial velocity of 20 feet per second. The height of the ball h in feet after t seconds is given by the equation h= -16t^2 - 20t + 240. How long after the ball is thrown will it strike the​ ground?

First check for the critical points of f by checking where the first-order derivatives vanish.

[tex]\dfrac{\partial f}{\partial x}=4x-4=0\implies x=1[/tex]

[tex]\dfrac{\partial f}{\partial y}=2y-4=0\implies y=2[/tex]

Notice how the point (1, 2) lies on the line y = 2x ; at this point, we get a value of f(1, 2) = -5 (MIN).

Next, check the points where the boundary lines intersect, which occurs at the points (0, 0), (0, 2), and (1, 2). We already checked the last one. We find f(0, 0) = 1 (MAX) and f(0, 2) = -3.

Now check on the boundary lines themselves. If x = 0, then

[tex]f(0,y)=y^2-4y+1=(y-2)^2-3[/tex]

which has a maximum value of -3 when y = 2 (so we get the same critical point as before).

If y = 2, then

[tex]f(x, 2)=2x^2-4x-3=2(x-1)^2-5[/tex]

with a maximum of -5 when x = 1.

If y = 2x, then

[tex]f(x,2x)=6x^2-12x+1=6(x-1)^2-5[/tex]

with the same maximum of -5 when x = 1.

This question is based on the absolute maximum and absolute minimum.

We get this by differentiating the terms.

Given:

f(x,y) =   [tex]2x^{2} - 4x + y^2 - 4y +1[/tex], bounded by the lines x=0,y=2,y=2x in the first quadrant,bounded by the lines x=0,y=2,y=2x in the first quadrant.

We need to determined the absolute maximum and absolute minimum of the function.

Now, partial differentiating wrt x and y.

[tex]\dfrac{\partial f}{ \partial x} = 4x -4 = 0 \Rightarrow x= 1 \\\dfrac{\partial f}{ \partial y} = 2y - 4 = 0 \Rightarrow y = 2[/tex]

Now, point (1, 2) lies on the line y = 2x ; at this point, we get a value of

f(1, 2) = -5 (MIN).

Next, check the points where the boundary lines intersect, which occurs at the points (0, 0), (0, 2), and (1, 2).

Now, find f(0, 0) = 1 (MAX) and f(0, 2) = -3.

Now check on the boundary lines themselves.

If x = 0, then we get,

[tex]f(0,y) = y^2 - 4y +1 = ( y-2)^2 -3\\[/tex]

which has a maximum value of -3 when y = 2 (so we get the same critical point as before).

If y = 2, then we get,

f(x,2) = [tex]2x^2-4x -3 = 2(x-1)^2 -5[/tex] with a maximum of -5 when x = 1.

If y = 2x, then we get,

f(x,2x) = [tex]6x^2 -12x +1 = 6(x-1)^2 -5[/tex] with the same maximum of -5 when     x = 1.

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Answer:

105.13ft^2

Step-by-step explanation:

[tex]A=lw\\=10*8\\=80ft^2[/tex]

Rectangle

[tex]A=\frac{1}{2} \pi r^2\\=\frac{1}{2\pi } 4^2\\=25.13[/tex]

Add both together

80+25.13

=105.13

Answer :  105.13

Step-by-step explanation:

Answer: Tiffany 15mph, Maggie 20mph

Step-by-step explanation:

Set up the equation 4((x+5) + x) = 140. x+5 represents how many miles Maggie covered in one hour. x represents how much Tiffany traveled in one hour. 140 is the number of miles in total. 4 is the number of hours in total.

Simplify the equation.

(x+5) + x = 35    Divide both sides by 4

2x+5 = 35         Combine like terms

2x = 30              Subtract 5 from both sides

x = 15                 Divide both sides by 2

Tiffany traveled 15mph, while Maggie traveled 15+5=20mph.

Answer:

Step-by-step explanation:

let s note a and b

x = ap+b

we can write two equations

(1) 300=3a+b

(2) 450=1.5a+b

multiply by 2 the (2) we got

900 =3a+2b

minus (1) it gives

900 - 300 = 3a+2b-3a-b = b

so b = 600

and from (1) it gives 3a = 300-600 = -300

so a = -100

then

x=-100p+600

thanks

Answer:

Step-by-step explanation:

[tex]a = 5 ,\\b = - 2 \\ c = - 2 \\ 2b - 3ac=?\\2(-2) -3(5)(-2)\\-4 +30\\= 26[/tex]

Answer:

-34

Step-by-step explanation:

2x-2-3(4)(-2)

which is -34

Answer:

226 cm^3

The mass of plastic used to make cylinder is greater

Step-by-step explanation:

Given:-

- The density of cone material, ρc = 1.4 g / cm^3

- The density of cylinder material, ρl = 0.8 g / cm^3

Solution:-

- To determine the volume of plastic that remains in the cylinder after gouging out a hemispherical amount of material.

- We will first consider a solid cylinder with length ( L = 10 cm ) and diameter ( d = 6 cm ). The volume of a cylinder is expressed as follows:

                                  [tex]V_L =\pi \frac{d^2}{4} * L[/tex]

- Determine the volume of complete cylindrical body as follows:

                                 [tex]V_L = \pi \frac{(6)^2}{4} * 10\\\\V_L = 90\pi cm^3\\[/tex]

- Where the volume of hemisphere with diameter ( d = 6 cm ) is given by:

                                 [tex]V_h = \frac{\pi }{12}*d^3[/tex]

- Determine the volume of hemisphere gouged out as follows:

                                 [tex]V_h = \frac{\pi }{12}*6^3\\\\V_h = 18\pi cm^3[/tex]

- Apply the principle of super-position and subtract the volume of hemisphere from the cylinder as follows to the nearest ( cm^3 ):

                               [tex]V = V_L - V_h\\\\V = 90\pi - 18\pi \\\\V = 226 cm^3[/tex]

Answer: The amount of volume that remains in the cylinder is 226 cm^3

- The volume of cone with base diameter ( d = 6 cm ) and height ( h = 5 cm ) is expressed as follows:

                               [tex]V_c = \frac{\pi }{12} *d^2 * h[/tex]

- Determine the volume of cone:

                              [tex]V_c = \frac{\pi }{12} *6^2 * 5\\\\V_c = 15\pi cm^3[/tex]

- The mass of plastic for the cylinder and the cone can be evaluated using their respective densities and volumes as follows:

                             [tex]m_i = p_i * V_i[/tex]

- The mass of plastic used to make the cylinder ( after removing hemispherical amount ) is:

                           [tex]m_L = p_L * V\\\\m_L = 0.8 * 226\\\\m_L = 180.8 g[/tex]

- Similarly the mass of plastic used to make the cone would be:

                           [tex]m_c = p_c * V_c\\\\m_c = 1.4 * 15\pi \\\\m_c = 65.973 g[/tex]

Answer: The total weight of the cylinder ( m_l = 180.8 g ) is greater than the total weight of the cone ( m_c = 66 g ).

The volume of the remaining plastic in the cylinder is large, which

makes the weight much larger than the weight of the cone.

Responses:

Density of the plastic for the cone = 1.4 g/cm³

Density of the plastic used for the cylinder = 0.8 g/cm³

From a similar question, we have;

Height of the cylinder = 10 cm

Diameter of the cylinder = 6 cm

Height of the cone = 5 cm

(a) Radius of the cylinder, r = 6 cm ÷ 2 = 3 cm

Volume of a cylinder = π·r²·h

Volume of a hemisphere = [tex]\mathbf{\frac{2}{3}}[/tex] × π×  r³

Volume of the cylinder after it has been hollowed out, V, is therefore;

Which gives;

[tex]V = \pi \times 3^2 \times 10 - \frac{2}{3} \times \pi \times 3^3 \approx \mathbf{ 226}[/tex]

(b) Volume of the cone = [tex]\mathbf{\frac{1}{3}}[/tex] × π × 3² × 5 ≈ 47.1

Mass of the cone = 47.1 cm³ × 1.4 g/cm³ ≈ 66 g

Mass of the hollowed cylinder ≈ 226 cm³ × 0.8 g/cm³ = 180.8 g

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Answer:

80

Step-by-step explanation:

17+13=30

22+23+5=50

30+50=80

Answer:

80 cards

Step-by-step explanation:

23+5=28

13+17=30

28+30=58

58+22=80

Triangle ABC was rotated 180° about the origin, then a by a scale factor of 1/3 was done to form triangle A'B'C'.

Transformation is the movement of a point from its initial location to a new location. Types of transformation are rotation, translation, reflection and dilation.

Given that;

Triangle ABC is similar to A"B"C".

Now, If a point A(x, y) is rotated clockwise by 180 degrees, the new point is at A'(y, -x)

Hence, Triangle ABC was rotated 180° about the origin, then a by a scale factor of 1/3 was done to form triangle A'B'C'.

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Answer:

[tex]\fbox{\begin{minipage}{8em}Not a solution\end{minipage}}[/tex]

Step-by-step explanation:

Step 1: Consider the assumption:

Generally, [tex](6, 7)[/tex]) is supposed to be the pair of 2 components, in which, the first component is x-component (domain), the second component is y-component (range).

Hence, [tex]x = 6, y = 7[/tex]

Step 2: Substitute [tex]x[/tex] and [tex]y[/tex] into the inequality

       [tex]y > 2x - 5[/tex]

<=>  [tex]7 > 2*6 - 5[/tex]

Step 3: Simplify

<=>  [tex]7> 12 - 5[/tex]

<=>  [tex]7 > 7[/tex]

Step 4: Evaluate

Invalid

Reason: [tex]7 = 7[/tex]

Step 5: Conclude

[tex](6, 7)[/tex] is not a solution to the inequality [tex]y > 2x - 5[/tex]

Hope this helps!

:)

Step-by-step explanation:

Simplifying

f(x) = 2cos(3x)

Multiply f * x

fx = 2cos(3x)

Remove parenthesis around (3x)

fx = 2cos * 3x

Reorder the terms for easier multiplication:

fx = 2 * 3cos * x

Multiply 2 * 3

fx = 6cos * x

Multiply cos * x

fx = 6cosx

Solving

fx = 6cosx

Solving for variable 'f'.

Move all terms containing f to the left, all other terms to the right.

Divide each side by 'x'.

f = 6cos

Simplifying

f = 6cos

Answer:

Step-by-step explanation:

20 sin^4x

=5(4sin^4 x)

=5(2sin²x)²

=5(1-cos 2x)²

=5(1-4cos2x+cos²(2x))

=5[1-4cos(2x)+{1+cos (4x)}/2]

=5/2[2-8cos(2x)+1+cos(4x)]

=5/2[3-8cos (2x)+cos (4x)]

Answer:

Step-by-step explanation:

To find the right answer, we just need to replace the given root in each expression and see which one gives zero.

[tex]f(x)=4x^{4} -7x^{2} +x+25\\f(-\frac{2}{5})= 4(-\frac{2}{5})^{4} -7(-\frac{2}{5})^{2} +(-\frac{2}{5})+25=\frac{64}{625}-\frac{28}{25} -\frac{2}{5} +25 \approx 23.58[/tex]

[tex]f(x)=9x^{4}-7x^{2} +x+10=9(-\frac{2}{5} )^{4} -7(-\frac{2}{5} )^{2} +\frac{2}{5} +10 \approx 9.5[/tex]

[tex]f(x)=10x^{4}-7x^{2} +x+9=10(-\frac{2}{5} )^{4} -7(-\frac{2}{5})^{2} +(-\frac{2}{5})+9 \approx 7.7[/tex]

[tex]f(x)=25x^{4}-7x^{2} +x+4=25(-\frac{2}{5} )^{4} -7(-\frac{2}{5})^{2} +(-\frac{2}{5})+4 \approx 3.12[/tex]

Therefore, neither expression satisfies the given rational root.

Answer:

D. f(x) = 25x^4 - 7x^2 + x + 4.

Step-by-step explanation:

The correct answer to your question is D.

Answer:

57°

Step-by-step explanation:

According to theorem, "any two angles in the same segment of the circle are equal"

So,

m<BED = 57°

Answer:

100+70+8+0.2+0.05. is the answer

Answer:

178.25 as a fraction is 178 1/4 or 713 / 4

Step-by-step explanation:

hope it works out !!

The correct answer is C. Mark’s method is correct because even though it is impossible to deliver mail to 1.2 houses in a minute, 1.2 represents the unit rate of houses per minute.

Explanation:

To begin Mark should determine the rate of delivery (number of houses the carrier can deliver in 1 minute). This can be found by dividing the houses by the minutes (36 / 30 = 1.2 houses per minute). This means the 1.2 rate found by Mark is correct; also, in this case, it is important to clarify, the carrier will not deliver to 1.2 houses at the same time, but this is the delivery rate or number used to understand the relationship between the number of houses, and the time.

Moreover, you can use this rate, and multiply it by 7.5 and this will show you how many houses the carrier can deliver in this time (7.5 (minutes) x 1.2 (delivery rate) = 9 houses). Thus, the method is correct, and in it, 1.2 represents the unit rate, this is why even when it is not possible to deliver to 1.2 houses all the process is correct.

Answer:

c

Step-by-step explanation:

i took the test

Answer:

The answer is C :,)

Step-by-step explanation:

Answer:

The answer to your question is c

Step-by-step explanation:

Because the years have to increase for it to grow.

Answer:

The sample size required is 289.

Step-by-step explanation:

Let p be population proportion of people that would buy the product.

It is provided that the nationwide poll on this type of product and price was run earlier this year, with percentages running from 75% to 80%.

Assume that the sample proportion of people that would buy the product is, [tex]\hat p=0.75[/tex].

A 95% Confidence Interval is to be constructed with a margin of error of 5%.

We need to determine the sample size required for the 95% Confidence Interval to be within 5% of the actual value.

The formula to compute the margin of error for a (1 - α)% confidence interval of population proportion is:

[tex]MOE=z_{\alpha/2}\times\sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]

The critical value of z for 95% confidence interval is,

z = 1.96.

Compute the sample size required as follows:

[tex]MOE=z_{\alpha/2}\times\sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]

      [tex]n=[\frac{z_{\alpha/2}\ \sqrt{\hat p(1-\hat p)} }{MOE}]^{2}[/tex]

         [tex]=[\frac{1.96\cdot \sqrt{0.75(1-0.75)} }{0.05}]^{2}\\\\=(16.9741)^{2}\\\\=288.12007081\\\\\approx 289[/tex]

Thus, the sample size required is 289.

Answer:

Laniece had 60 magazines

Step-by-step explanation:

Given: In the  first week, Tamia sold 30% more than Laniece. In the second  week, Tamia sold 12 magazines, but Laniece did not sell any. Tamia sold 50% more than Laniece by the end of the second  week

To find: Number of magazines sold by Laniece

Solution:

Let number of magazines sold by Laniece in the first week be x.

Number of magazines sold by Tamia in the first week = [tex]x+\frac{30}{100} x=\frac{130x}{100} =\frac{13x}{10}[/tex]

Number of magazines sold by Tamia in the second week = 12

Total number of magazines sold by Tamia at the end of the second week = [tex]\frac{13x}{10}+12[/tex]

Total number of magazines sold by Laniece at the end of the second week = x

According to question,

[tex]\frac{13x}{10}+12=x+\frac{50x}{100}=x+\frac{x}{2}\\\frac{13x}{10}+12=\frac{3x}{2}\\\frac{3x}{2}-\frac{13x}{10} =12\\\frac{15x-13x}{10}=12\\\frac{2x}{10}=12\\\frac{x}{5}=12\\x=60[/tex]

Answer:

1607.68 square miles

Step-by-step explanation:

use pi r squared

Answer:

Step-by-step explanation:

diameter = 64  miles

r =64/2 = 32 miles

Area of semicircle = πr²/2

                              = 3.14*32*32/2

                             = 1607.68 sq.miles

Answer:

Please see steps below

Step-by-step explanation:

In order to factor by grouping, we divide the four terms given into two groups, and extract on each group any common factor  we can.

In our example, we can select the terms: [tex]w^2[/tex]  and [tex]w[/tex] as one of our groups, and [tex]3w[/tex] and 3 in the other group. Then we re-organize the expression as:

[tex](w^2+w)+(3x+3)[/tex]

Now we extract from the first binomial group, the factor [tex]w[/tex] as a common factor of both terms, and from the second group we extract the factor "3" as common factor of those two terms:

[tex](w^2+w)+(3x+3)\\w(w+1)+3(w+1)[/tex]

We notice now that after the extraction, we are left with two exactly equal binomial factors [tex](w+1)[/tex] that appeared in the first group and in the second group. We proceed then to extract it as common factor for the two groups:

[tex]w(w+1)+3(w+1)\\(w+1)(w+3)[/tex]

this last product of two binomials ([tex](w+1)\,(w+3)[/tex] is the result of factoring the original expression.

Answer:

b. The engineer who weighed the rod 25 times.

Step-by-step explanation:

Hello!

Full text:

Length of a rod: Engineers on the Bay Bridge are measuring tower rods to find out if any rods have been corroded from salt water. There are rods on the east and west sides of the bridge span. One engineer plans to measure the length of an eastern rod 25 times and then calculate the average of the 25 measurements to estimate the true length of the eastern rod. A different engineer plans to measure the length of a western rod 20 times and then calculate the average of the 20 measurements to estimate the true length of the western rod.

Suppose the engineers construct a 90% confidence interval for the true length of their rods. Whose interval do you expect to be more precise (narrower)?

a. Both confidence intervals would be equally precise.

b. The engineer who weighed the rod 25 times.

c. The engineer who weighed the rod 20 times.

X₁: Length of an eastern rod of the Bay Bridge

n₁= 25

X₂: Length of a western rod of the Bay Bridge

n₂= 20

Both Engineers will use their samples to estimate the population average length of the rods using a 90% CI.

Assuming the standard normal distribution, the confidence interval will be centered in the estimated mean.

X[bar] ± [tex]Z_{1-\alpha /2}[/tex]*(σ/√n)

And the width is determined by the semi amplitude:

↓d= [tex]Z_{1-\alpha /2}[/tex]*(σ/√↑n)

As you can see the sample size has an indirect relationship with the semi amplitude of the interval. This means, when the sample size increases, the semi amplitude decreases, and if the sample size decreases, the semi amplitude increases. Naturally this is leaving all other elements of the equation constant, this means, using the same confidence level and the same population standard deviation.

Since the first engineer took the larger sample, he's confidence interval will be narrower and more accurate.

Hope this helps!

Answer:

0.0004% probability that the student answers at least 9 questions correctly

Step-by-step explanation:

For each question, there are only two possible outcomes. Either the student guesses the correct answer, or he does not. All questions are answered independently. This means that we use the binomial distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

And p is the probability of X happening.

In this question, we have that:

[tex]n = 10, p = 0.2[/tex]

What is the probability that the student answers at least 9 questions correctly

[tex]P(X \geq 9) = P(X = 9) + P(X = 10)[/tex]

In which

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 9) = C_{10,9}.(0.2)^{9}.(0.8)^{1} = 0.000004[/tex]

[tex]P(X = 10) = C_{10,10}.(0.2)^{10}.(0.8)^{0} \approx 0 [/tex]

[tex]P(X \geq 9) = P(X = 9) + P(X = 10) = 0.000004 + 0 = 0.000004[/tex]

0.0004% probability that the student answers at least 9 questions correctly

Answer:

48x>+2

Step-by-step explanation:

Answer:

height = 6 mm

Step-by-step explanation:

The prism is a rectangular prism. The base area of the prism is 24 mm². The volume of the prism is given as 144 mm³.

The height of the prism can be solved as follows.

Volume of the rectangular prism = Bh

where

B = base area

h = height

Volume = 144 mm³

B = 24 mm²

volume = Bh

144 = 24 × h

144 = 24h

divide both sides by 24

h = 144/24

h = 6 mm

Answer:

c

Step-by-step explanation:

edg 2022

Answer:

B and F

Step-by-step explanation:

When we'll slice a triangular prism, a square and triangle would be formed

Answer:

8

Step-by-step explanation:

y²+by+16= (y+4)²

y²+by+16= y²+2*4*y+4²

y²+by+16= y²+8y+16

by=8y

b=8

Answer:

Step-by-step explanation:

From the question; we are given the following inclusive frequency distribution information

Class                Frequency f

1.1-1.5                           18

1.6-2.0                         27

2.1-2.5                         31

2.6-3.0                        40

3.1-3.5                         56

3.6-4.0                        55

4.1-4.5                          23

Convert the above inclusive frequency distribution to exclusive frequency distribution with respect of the upper and lower class limit ; we have:

Class                             Frequency f

1.05 - 1.55                             18

1.55 - 2.05                           27

2.05 - 2.55                          31

2.55 - 3.05                          40

3.05 - 3.55                          56

3.55 - 4.05                          55

4.05 - 4.55                          23

Class                             Frequency f                     cf

1.05 - 1.55                             18                              18            

1.55 - 2.05                           27                               45

2.05 - 2.55                          31                               76

2.55 - 3.05                          40                               116

3.05 - 3.55                          56                               172

3.55 - 4.05                          55                               227

4.05 - 4.55                          23                               250

                                       n = 250

To determine the daily sales; we can derive that from estimated Mode by using the relation :

Estimated Mode = L + fm − fm-1(fm − fm-1) + (fm − fm+1) × w

here:

L  = the lower class boundary of the modal group

fm-1 =  the frequency of the group before the modal group

fm = the frequency of the modal group

fm+1 = the frequency of the group after the modal group

w  = the group width

However;

It is easier now to determine  the modal group (i.e the group with the highest frequency), which is 3.05 -3.55

L = 3.05

fm-1 =40

fm =56

fm+1 = 55

w = 0.5

∴[tex]mode = 3.05 + \dfrac{56 - 40 }{(56 - 40) + (56 -55)} * 0.5 \\ \\ mode = 3.05 + 0.4705 \\ \\ mode = 3.5205[/tex]

To find Median Class ; we use the formula;

Median Class = value of (n / 2)th observation

Median Class  = value of (250 / 2)th observation  

Median Class = value of 125th observation

From the column of cumulative frequency cf,

we will see that the 125th observation lies in the class 3.05-3.55.

∴ The median class is 3.05-3.55.

Thus;,  

L=lower boundary point of median class =3.05

n=Total frequency =250

cf=Cumulative frequency of the class preceding the median class =116

f=Frequency of the median class =56

c=class length of median class =0.5

[tex]Median M=L+n2-cff- c \\ \\ =3.05+125-11656⋅0.5 \\ \\=3.05+0.08036 \\ \\ =3.13036[/tex]

hence median sales = $3130.36