(100 pts.)

levacccp [35]

A and b are not true because

7 0

3 years ago

Find the area of the region bounded by the parabola y = 2x^2, the tangent line to this parabola at (4, 32), and the x-axis.

Anna [14]

I have a solution here that has a slight change in given where instead of <span>(4, 32), it is (3, 18). However, since the solution has provided explanations on each process, step-by-step, I believe that by thoroughly analyzing it, you might just answer this problem on your own!
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f(x) = 2x² ← this is the parabola

f(3) = 2 * 9 = 18 → the parabola passes through A (3 ; 18), so its tangent line too

f'(x) = 4x ← this is the derivative

…and the derivative is the slope of the tangent line to the curve at x

f'(3) = 4 * 3 = 12 ← this is the slope of the tangent line to the curve at x = 3

Equation of the tangent line

The typical equation of a line is: y = mx + b → where m: slope and where b: y-intercept

You know that the slope of the tangent line is 12.

The equation of the tangent line becomes: y = 12x + b

The tangent line passes through A (3 ; 18), so these coordinates must verify the equation of the tangent line.

y = 12x + b

b = y - 12x → you substitute x and y by the coordinates of the point A (3 ; 18)

b = 18 - 36 = - 18

→ The equation of the tangent line is: y = 12x - 18

Intersection between the tangent line to the curve and the x-axis: → when y = 0

y = 12x - 18 → when y = 0

12x - 18 = 0

12x = 18

x = 3/2

→ Point B (3/2 ; 0)

Intersection between the vertical line passes through the point A and the x-axis: → when x = 3

→ Point C (3 ; 0)

The equation of the vertical line is: x = 3

Area of the region bounded by the parabola y = 2x², the tangent line to this parabola at (3 ; 18), and the x-axis.

= (area of the region bounded by the parabola y = 2x² and the x-axis) - (area of the triangle ABC)

= [∫ (from 0 to 3) of the parabola] - [(xC - xB).(yA - yC)/2]

= [∫ (from 0 to 3) 2x².dx] - [(xC - xB).(yA - yC)/2]

= { [(2/3).x³] from 0 to 3 } - { [3 - (3/2)].(18 - 0)/2 }

= [(2/3) * 3³] - { [(6/2) - (3/2)] * 9 }

= [(2/3) * 27] - { [(3/2) * 9 }

= 18 - (27/2)

= (36/2) - (27/2)

= 9/2 square unit

6 0

3 years ago

What is the solution to the following equation?

Murrr4er [49]

Answer:

I think the answer is letter A.

6 0

3 years ago

A random sample of 500 adult residents of Maricopa County found that 356 were in favor of increasing the highway speed limit to

zzz [600]

Answer:

$z=1.04$

$p_v =2*P(Z>1.04)\approx 0.298$

Comparing the p value with the significance level given $\alpha=0.05$ we see that $p_v>\alpha$ so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can't say that the proportion of people in favor of the increased speed limit not differs significantly between the two groups.

Step-by-step explanation:

1) Data given and notation

$X_{PC}=297$ represent the number of residents of Pima County in favor of the increased speed limit

$X_{MC}=356$ represent the number of residents of Maricopa County in favor of the increased speed limit

$n_{PC}=400$ sample of Pima County selected

$n_{MC}=500$ sample of Maricopa County selected

$p_{PC}=\frac{297}{400}=0.743$ represent the proportion of residents of Pima County in favor of the increased speed limit

$p_{MC}=\frac{356}{500}=0.712$ represent the proportion of residents of Maricopa County in favor of the increased speed limit

z would represent the statistic (variable of interest)

$p_v$ represent the value for the test (variable of interest)

$\alpha=0.05$ significance level given

2) Concepts and formulas to use

We need to conduct a hypothesis in order to check if is there is a difference in the support for in increasing the speed limit between the residents of the two counties, the system of hypothesis would be:

Null hypothesis: $p_{PC} - p_{MC}=0$

Alternative hypothesis: $p_{PC} - \mu_{MC} \neq 0$

We need to apply a z test to compare proportions, and the statistic is given by:

$z=\frac{p_{PC}-p_{MC}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{PC}}+\frac{1}{n_{MC}})}}$ (1)

Where $\hat p=\frac{X_{PC}+X_{MC}}{n_{PC}+n_{MC}}=\frac{297+356}{400+500}=0.726$

z-test: Is used to compare group means. Is one of the most common tests and is used to determine whether the means of two groups are equal to each other.

3) Calculate the statistic

Replacing in formula (1) the values obtained we got this:

$z=\frac{0.743-0.712}{\sqrt{0.726(1-0.726)(\frac{1}{400}+\frac{1}{500})}}=1.04$

4) Statistical decision

Since is a two side test the p value would be:

$p_v =2*P(Z>1.04)\approx 0.298$

Comparing the p value with the significance level given $\alpha=0.05$ we see that $p_v>\alpha$ so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can't say that the proportion of people in favor of the increased speed limit not differs significantly between the two groups.

8 0

3 years ago

Ben the camel drinks tea (so classy!). He drinks 350 liters of tea every 2 days.

Eddi Din [679]

Every six days he'd drink 1,050 liters because

350 liters every 2 days -

350 x 2 = 700

700 liters every 4 days -

2 x 3 = 6

1,050 liters every 6 days.

Hope this helps.

4 0

3 years ago

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