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4 years ago

Find the area of \triangle ABC△ABCtriangle, A, B, C.

Mathematics

2 answers:

Bezzdna [24]4 years ago

7 0

Answer:

6.0square units

Step-by-step explanation:

Area of a triangle = 1/2 × base × height

The height of the given triangle is CD and the base is AB

To get the height CD, we need to use the Pythagoras theorem on the triangle ACD

AC² = CD² + AD²

CD² = 3²-1.8²

CD² = 5.76

CD = √5.76

CD = 2.4

According to ∆ABC,

AB² = AC²+BC²

Since AB = AD+DB then;

(AD+DB)² = AC²+BC²

AD² + 2(AD)(DB)+DB² = AC²+BC²

DB² = AC²+BC² - AD² - 2(AD)(DB)... 1

According to ∆BCD

BC² = CD²+DB²

DB² = BC²-CD²...2

Equating 1 and 2

AC²+BC² - AD² - 2(AD)(DB) = BC²-CD²

AC² - AD² - 2(AD)(DB) + CD² = 0

AC = 3

AD = 1.8

CD = 2.4

On substituting

3²-1.8² - 2(1.8)DB +2.4² = 0

9-3.24-3.6DB+5.76 = 0

11.52 - 3.6DB = 0

3.6DB = 11.52

DB = 11.52/3.6

DB = 3.2

The base of the triangle

AB = AD+DB

AB = 1.8+3.2

AB = 5.0

Area of the triangle = 1/2 × 2.4 × 5

= 1.2 × 5

= 6.0square units

Sergio [31]4 years ago

4 0

Answer:

= 6 square unit

Step-by-step explanation:

Area of a triangle is 1/2 × base × height

first we find CD

we use Pythagoras theorem

we represent CD with x

3² = 1.8² + x²

9 = 3.24 + x²

x² = 9 - 3.24

x² = 5.76

x = √5.76

x = 2.4

CD = 2.4

Area of a triangle = 1/2 × base × height

According to ∆ABC,

AB² = AC²+BC²

Since AB = AD+DB then;

(AD+DB)² = AC²+BC²

AD² + 2(AD)(DB)+DB² = AC²+BC²

DB² = AC²+BC² - AD² - 2(AD)(DB)... 1

According to ∆BCD

BC² = CD²+DB²

DB² = BC²-CD²...2

Equating 1 and 2

AC²+BC² - AD² - 2(AD)(DB) = BC²-CD²

AC² - AD² - 2(AD)(DB) + CD² = 0

AC = 3

AD = 1.8

CD = 2.4

On substituting

3²-1.8² - 2(1.8)DB +2.4² = 0

9-3.24-3.6DB+5.76 = 0

11.52 - 3.6DB = 0

3.6DB = 11.52

DB = 11.52/3.6

DB = 3.2

The base of the triangle

AB = AD+DB

AB = 1.8+3.2

AB = 5.0

Area of the triangle = 1/2 × 2.4 × 5

= 1.2 × 5

= 6 square unit

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3 years ago

Elderly drivers. In January 2011, The Marist Poll published a report stating that 66% of adults nationally think licensed driver

katrin [286]

Answer:

(a) Hence, the margin of error reported by The Marist Poll was correct.

(b) Based on a 95% confidence interval the poll does not provide convincing evidence that more than 70% of the population think that licensed drivers should be required to retake their road test once they turn 65.

Step-by-step explanation:

We are given that the Marist Poll published a report stating that 66% of adults nationally think licensed drivers should be required to retake their road test once they reach 65 years of age.

It was also reported that interviews were conducted on 1,018 American adults, and that the margin of error was 3% using a 95% confidence level.

(a) Margin of error formula is given by;

Margin of Error = $Z_(_\frac{\alpha}{2}_) \times \sqrt{\frac{\hat p(1-\hat p)}{n} }$

where, $\alpha$ = level of significance = 1 - 0.95 = 0.05 or 5%

Standard of error = $\sqrt{\frac{\hat p(1-\hat p)}{n} }$

Also, $\hat p$ = sample proportion of adults nationally think licensed drivers should be required to retake their road test once they reach 65 years of age = 66%

n = sample of American adults = 1.018

The critical value of z for level of significance of 2.5% is 1.96.

So, Margin of Error = $Z_(_\frac{\alpha}{2}_) \times \sqrt{\frac{\hat p(1-\hat p)}{n} }$

= $1.96 \times \sqrt{\frac{0.66(1-0.66)}{1,018} }$ = 0.03 or 3%

Hence, the margin of error reported by The Marist Poll was correct.

(b) Now, the pivotal quantity for 95% confidence interval for the population proportion who think that licensed drivers should be required to retake their road test once they turn 65 is given by;

P.Q. = $\frac{\hat p-p}{ \sqrt{\frac{\hat p(1-\hat p)}{n} }}$ ~ N(0,1)

So, 95% confidence interval for p = $\hat p \pm \text{Margin of error}$

= $0.66 \pm 0.03$

= [0.66 - 0.03 , 0.66 + 0.03]

= [0.63 , 0.69]

Hence, based on a 95% confidence interval the poll does not provide convincing evidence that more than 70% of the population think that licensed drivers should be required to retake their road test once they turn 65 because the interval does not include the value of 70% or more.

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