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10 days ago

In ΔHIJ, the measure of ∠J=90°, JI = 4, HJ = 3, and IH = 5. What ratio represents the tangent of ∠I?

Mathematics

1 answer:

Inessa [3.9K]10 days ago

7 0

Answer:

The ratio $\frac{3}{4}$ corresponds to the tangent of ∠I.

Step-by-step explanation:

Let’s revisit the trigonometric ratios:

sin Ф = $\frac{opposite}{hypotenuse}$
cos Ф = $\frac{adjacent}{hypotenuse}$
tan Ф = $\frac{opposite}{adjacent}$

For triangle HIJ

∵ m∠J = 90°

- The hypotenuse is the side opposite the right angle.

So, HI is the hypotenuse.

∵ HJ = 3 units

∵ IH = 5 units

- We’ll apply the Pythagorean Theorem to solve for HJ.

∵ (HJ)² + (IJ)² = (IH)²

∴ 3² + (IJ)² = 5²

∴ 9 + (IJ)² = 25

- Subtract 9 from both sides.

∴ (IJ)² = 16

- Taking the square root on both sides gives:

∴ IJ = 4 units

To determine the tangent of ∠I, identify the sides that are opposite and adjacent to it.

∵ HJ is opposite to ∠I

∵ IJ is adjacent to ∠I

- Utilizing the rule of tan above:

∴ tan(∠I) = $\frac{HJ}{IJ}$

∴ tan(∠I) = $\frac{3}{4}$

The ratio $\frac{3}{4}$ indicates the tangent of ∠I.

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Answer:

link:

Step-by-step explanation:

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8 days ago

Jacque needs to buy some pizzas for a party at her office. She's ordering from a restaurant that charges a \$7.50$7.50dollar sig

Svet_ta [4321]

Answer:

Step-by-step explanation:

Delivery charge = $7.50

Cost per pizza = $14

Budgeted total = $60

7.50 + 14p < 60

First, subtract 7.50 from both sides

14p < 60 - 7.50

14p < 52.50

Now divide both sides by 14

p < 52.50 / 14

p < 3.75

Each pizza consists of 8 slices

Therefore, the total number of slices she can buy = 3.75 × 8

= 30 slices

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17 days ago

Find the dimensions of the closed rectangular box with maximum volume that can be inscribed in the unit sphere.

Inessa [3907]

Answer:

Wyjaśnienie krok po kroku:

Załóżmy, że prostokąt ma wierzchołki (x,y,z) w dodatnim oktancie. Prostokątny pudełko musi być symetryczne względem wszystkich trzech osi.

Wtedy jego boki to

$2x,2y,2z$

Objętość = $8xyz$

Maksymalizuj objętość przy założeniu

$x^2+y^2+z^2 =1$

tj. $g(x,y,z) = x^2+y^2+z^2 -1=0$

Użyj mnożników Lagrange'a, mamy

$∇f(x,y,z)=λ∇g(x,y,z)$ przy maksimum

$∇f(x,y,z)=λ∇g(x,y,z) =(8yz,8xz,8xy)\\∇g(x,y,z)=(2x,2y,2z)$

$8yz=2λx8xz=2λy8xy=2λz$

Dzieląc otrzymujemy

$\frac{y}{x} =\frac{x}{y} \\x^2=y^2$

Podobnie $y^2=z^2$

Zatem otrzymujemy $3x^2 =1\\x = \frac{1}{\sqrt{3} }$

Stąd wymiary to

(2x,2y,2z)

$\frac{2}{\sqrt{3} },\frac{2}{\sqrt{3} },\frac{2}{\sqrt{3} } )$

</pzatem>

3 0

1 day ago

Use the normal approximation to the binomial distribution to answer this question. Fifteen percent of all students at a large un

AnnZ [3877]

Answer: 0.1289

Step-by-step explanation:

Given: The proportion of students absent on Mondays at a large university.: $p=0.15$

Sample size: $n=12$

Mean: $\mu=np=12\times0.15=1.8$

Standard deviation = $\sigma=\sqrt{np(1-p)}$

$\Rightarrow\ \sigma=\sqrt{12(0.15)(1-0.15)}=1.23693168769\approx1.2369$

Let x represent a binomial variable.

Referencing the standard normal distribution table,

$P(x=4)=P(x\leq4)-P(x\leq3)$ (1)

Z score for normal distribution:-

$z=\dfrac{x-\mu}{\sigma}$

For x=4

$z=\dfrac{4-1.8}{1.2369}\approx1.78$

For x=3

$z=\dfrac{3-1.8}{1.2369}\approx0.97$

Thus, from (1)

$P(x=4)=P(z\leq1.78)-P(z\leq0.97)\\\\=0.962462-0.8339768\approx0.1289$

Consequently, the likelihood of four students being absent = $0.1289$

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16 hours ago

A laundry basket contains 18 blue socks and 24 black socks. What is the probability of randomly picking 2 black socks, without r

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<span>P(black socks): 24/42 or 12/21
P(black socks without replacement): 23/41. Consequently, the chance of randomly selecting 2 black socks, without replacement, from the basket is 12/21×23/41=276/861, equivalent to 32%. Hope this helps!</span>

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2 days ago