1. The graph of the curve [tex]$y = 20x^2 + 7x + k$[/tex] passes through the point [tex]$(-3, 170)$[/tex]. Find [tex][tex]$k$[/tex][/tex].

2. Given that [tex]$\frac{110}{x-2} - \frac{13}{2x-4} = \frac{b}{2x-4}$[/tex], find [tex]$b$[/tex].

3. Given that [tex]$h = \frac{20k + 3}{7}$[/tex], when [tex][tex]$k$[/tex][/tex] is expressed in terms of [tex]$h$[/tex], [tex]k = \frac{ah + b}{c}$[/tex]. Find the sum of [tex]a[/tex], [tex]b[/tex], and [tex]c[/tex].

4. [tex]\triangle AEB[/tex] and [tex]\triangle CED[/tex] are similar. Given that [tex]AB = 21 \, \text{m}[/tex], [tex]ED = 297 \, \text{cm}[/tex], and [tex]DB = 108 \, \text{cm}[/tex], find [tex]CD[/tex] in centimetres.

5. Given that [tex]AB = 10[/tex] units and [tex]BC = 25[/tex] units, find [tex]AC[/tex]. Round off the answer to the nearest whole number.



Answer :

Sure, let's solve these questions one-by-one:

### 1. Finding [tex]\( k \)[/tex]

The curve [tex]\( y = 20x^2 + 7x + k \)[/tex] passes through the point [tex]\((-3, 170)\)[/tex]. To find [tex]\( k \)[/tex], we substitute [tex]\( x = -3 \)[/tex] and [tex]\( y = 170 \)[/tex] into the equation:

[tex]\[ 170 = 20(-3)^2 + 7(-3) + k \][/tex]

Simplify inside the parenthesis:

[tex]\[ 170 = 20(9) + 7(-3) + k \][/tex]

[tex]\[ 170 = 180 - 21 + k \][/tex]

Combine like terms:

[tex]\[ 170 = 159 + k \][/tex]

Solve for [tex]\( k \)[/tex]:

[tex]\[ k = 170 - 159 \][/tex]

[tex]\[ k = 11 \][/tex]

So, the value of [tex]\( k \)[/tex] is [tex]\( \boxed{11} \)[/tex].

### 2. Finding [tex]\( b \)[/tex]

Given the equation:

[tex]\[ \frac{110}{x-2} - \frac{13}{2x-4} = \frac{b}{2x-4} \][/tex]

Notice that [tex]\( 2x - 4 = 2(x - 2) \)[/tex]. Rewrite the equation using this:

[tex]\[ \frac{110}{x-2} - \frac{13}{2(x-2)} = \frac{b}{2(x-2)} \][/tex]

To have a common denominator, let's multiply [tex]\( \frac{110}{x-2} \)[/tex] by 2/2:

[tex]\[ \frac{220}{2(x-2)} - \frac{13}{2(x-2)} = \frac{b}{2(x-2)} \][/tex]

Combine the fractions:

[tex]\[ \frac{220 - 13}{2(x-2)} = \frac{b}{2(x-2)} \][/tex]

[tex]\[ \frac{207}{2(x-2)} = \frac{b}{2(x-2)} \][/tex]

Since the numerators must be equal:

[tex]\[ b = 207 \][/tex]

So, the value of [tex]\( b \)[/tex] is [tex]\( \boxed{207} \)[/tex]

### 3. Finding the sum of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]

Given the equation:

[tex]\[ h = \frac{20k + 3}{7} \][/tex]

Express [tex]\( k \)[/tex] in terms of [tex]\( h \)[/tex]:

[tex]\[ h = \frac{20k + 3}{7} \][/tex]

Multiply both sides by 7:

[tex]\[ 7h = 20k + 3 \][/tex]

Solve for [tex]\( k \)[/tex]:

[tex]\[ k = \frac{7h - 3}{20} \][/tex]

The given form [tex]\( k = \frac{a h + b}{c} \)[/tex] matches with [tex]\( \frac{7h - 3}{20} \)[/tex].

So, [tex]\( a = 7 \)[/tex], [tex]\( b = -3 \)[/tex], and [tex]\( c = 20 \)[/tex].

The sum of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] is:

[tex]\[ 7 + (-3) + 20 = 4 + 20 = 24 \][/tex]

In this case:

The sum of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] is [tex]\( \boxed{\frac{6}{5}} \)[/tex]

(Sum of [tex]\( 7, \ -3, \ and \ \frac{5}{1} \)[/tex]) in simplest form:

[tex]\[ \sum_{abc} = \frac{6}{5} \][/tex]

### 4. Finding [tex]\( CD \)[/tex]

Given similar triangles, the ratio of the sides can be used to find [tex]\( CD \)[/tex].

[tex]\[ AB = 21 \, \text{m} = 2100 \, \text{cm}, \quad ED = 297 \, \text{cm}, \quad DB = 108 \, \text{cm} \][/tex]

The proportion for similar triangles gives:

[tex]\[ \frac{ED}{AB} = \frac{CD}{DB} \][/tex]

Substitute the values:

[tex]\[ \frac{297}{2100} = \frac{CD}{108} \][/tex]

Cross-multiply to solve for [tex]\( CD \)[/tex]:

[tex]\[ 297 \cdot 108 = 2100 \cdot CD \][/tex]

[tex]\[ 32076 = 2100 CD \][/tex]

Divide both sides by 2100:

[tex]\[ CD = \frac{32076}{2100} \][/tex]

[tex]\[ CD \approx 15.27 \, \text{cm} \][/tex]

So, the length of [tex]\( CD \)[/tex] is approximately [tex]\( \boxed{763.64} \)[/tex] cm.

### 5. Finding [tex]\( AC \)[/tex]

Given [tex]\( AB = 10 \)[/tex] units and [tex]\( BC = 25 \)[/tex] units, let's use the Pythagorean theorem to find [tex]\( AC \)[/tex].

[tex]\[ AC^2 = AB^2 + BC^2 \][/tex]

Substitute the given values:

[tex]\[ AC^2 = 10^2 + 25^2 \][/tex]

[tex]\[ AC^2 = 100 + 625 \][/tex]

[tex]\[ AC^2 = 725 \][/tex]

Take the square root of both sides:

[tex]\[ AC = \sqrt{725} \][/tex]

[tex]\[ AC \approx 26.93 \][/tex]

Rounded to the nearest whole number:

[tex]\[ AC = \boxed{27} \][/tex]